Squashed 'third_party/eigen/' changes from 61d72f6..cf794d3
Change-Id: I9b814151b01f49af6337a8605d0c42a3a1ed4c72
git-subtree-dir: third_party/eigen
git-subtree-split: cf794d3b741a6278df169e58461f8529f43bce5d
diff --git a/unsupported/Eigen/src/MatrixFunctions/CMakeLists.txt b/unsupported/Eigen/src/MatrixFunctions/CMakeLists.txt
deleted file mode 100644
index cdde64d..0000000
--- a/unsupported/Eigen/src/MatrixFunctions/CMakeLists.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-FILE(GLOB Eigen_MatrixFunctions_SRCS "*.h")
-
-INSTALL(FILES
- ${Eigen_MatrixFunctions_SRCS}
- DESTINATION ${INCLUDE_INSTALL_DIR}/unsupported/Eigen/src/MatrixFunctions COMPONENT Devel
- )
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
index 6825a78..e5ebbcf 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
@@ -1,8 +1,8 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
-// Copyright (C) 2009, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
-// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
+// Copyright (C) 2009, 2010, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2011, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
@@ -14,376 +14,374 @@
#include "StemFunction.h"
namespace Eigen {
+namespace internal {
-/** \ingroup MatrixFunctions_Module
- * \brief Class for computing the matrix exponential.
- * \tparam MatrixType type of the argument of the exponential,
- * expected to be an instantiation of the Matrix class template.
- */
-template <typename MatrixType>
-class MatrixExponential {
+/** \brief Scaling operator.
+ *
+ * This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
+ */
+template <typename RealScalar>
+struct MatrixExponentialScalingOp
+{
+ /** \brief Constructor.
+ *
+ * \param[in] squarings The integer \f$ s \f$ in this document.
+ */
+ MatrixExponentialScalingOp(int squarings) : m_squarings(squarings) { }
- public:
- /** \brief Constructor.
- *
- * The class stores a reference to \p M, so it should not be
- * changed (or destroyed) before compute() is called.
- *
- * \param[in] M matrix whose exponential is to be computed.
- */
- MatrixExponential(const MatrixType &M);
+ /** \brief Scale a matrix coefficient.
+ *
+ * \param[in,out] x The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
+ */
+ inline const RealScalar operator() (const RealScalar& x) const
+ {
+ using std::ldexp;
+ return ldexp(x, -m_squarings);
+ }
- /** \brief Computes the matrix exponential.
- *
- * \param[out] result the matrix exponential of \p M in the constructor.
- */
- template <typename ResultType>
- void compute(ResultType &result);
+ typedef std::complex<RealScalar> ComplexScalar;
+
+ /** \brief Scale a matrix coefficient.
+ *
+ * \param[in,out] x The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
+ */
+ inline const ComplexScalar operator() (const ComplexScalar& x) const
+ {
+ using std::ldexp;
+ return ComplexScalar(ldexp(x.real(), -m_squarings), ldexp(x.imag(), -m_squarings));
+ }
private:
-
- // Prevent copying
- MatrixExponential(const MatrixExponential&);
- MatrixExponential& operator=(const MatrixExponential&);
-
- /** \brief Compute the (3,3)-Padé approximant to the exponential.
- *
- * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
- * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
- *
- * \param[in] A Argument of matrix exponential
- */
- void pade3(const MatrixType &A);
-
- /** \brief Compute the (5,5)-Padé approximant to the exponential.
- *
- * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
- * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
- *
- * \param[in] A Argument of matrix exponential
- */
- void pade5(const MatrixType &A);
-
- /** \brief Compute the (7,7)-Padé approximant to the exponential.
- *
- * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
- * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
- *
- * \param[in] A Argument of matrix exponential
- */
- void pade7(const MatrixType &A);
-
- /** \brief Compute the (9,9)-Padé approximant to the exponential.
- *
- * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
- * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
- *
- * \param[in] A Argument of matrix exponential
- */
- void pade9(const MatrixType &A);
-
- /** \brief Compute the (13,13)-Padé approximant to the exponential.
- *
- * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
- * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
- *
- * \param[in] A Argument of matrix exponential
- */
- void pade13(const MatrixType &A);
-
- /** \brief Compute the (17,17)-Padé approximant to the exponential.
- *
- * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
- * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
- *
- * This function activates only if your long double is double-double or quadruple.
- *
- * \param[in] A Argument of matrix exponential
- */
- void pade17(const MatrixType &A);
-
- /** \brief Compute Padé approximant to the exponential.
- *
- * Computes \c m_U, \c m_V and \c m_squarings such that
- * \f$ (V+U)(V-U)^{-1} \f$ is a Padé of
- * \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. The
- * degree of the Padé approximant and the value of
- * squarings are chosen such that the approximation error is no
- * more than the round-off error.
- *
- * The argument of this function should correspond with the (real
- * part of) the entries of \c m_M. It is used to select the
- * correct implementation using overloading.
- */
- void computeUV(double);
-
- /** \brief Compute Padé approximant to the exponential.
- *
- * \sa computeUV(double);
- */
- void computeUV(float);
-
- /** \brief Compute Padé approximant to the exponential.
- *
- * \sa computeUV(double);
- */
- void computeUV(long double);
-
- typedef typename internal::traits<MatrixType>::Scalar Scalar;
- typedef typename NumTraits<Scalar>::Real RealScalar;
- typedef typename std::complex<RealScalar> ComplexScalar;
-
- /** \brief Reference to matrix whose exponential is to be computed. */
- typename internal::nested<MatrixType>::type m_M;
-
- /** \brief Odd-degree terms in numerator of Padé approximant. */
- MatrixType m_U;
-
- /** \brief Even-degree terms in numerator of Padé approximant. */
- MatrixType m_V;
-
- /** \brief Used for temporary storage. */
- MatrixType m_tmp1;
-
- /** \brief Used for temporary storage. */
- MatrixType m_tmp2;
-
- /** \brief Identity matrix of the same size as \c m_M. */
- MatrixType m_Id;
-
- /** \brief Number of squarings required in the last step. */
int m_squarings;
+};
- /** \brief L1 norm of m_M. */
- RealScalar m_l1norm;
+/** \brief Compute the (3,3)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade3(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatA>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {120.L, 60.L, 12.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (5,5)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade5(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {30240.L, 15120.L, 3360.L, 420.L, 30.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (7,7)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade7(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {17297280.L, 8648640.L, 1995840.L, 277200.L, 25200.L, 1512.L, 56.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2
+ + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+
+}
+
+/** \brief Compute the (9,9)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade9(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {17643225600.L, 8821612800.L, 2075673600.L, 302702400.L, 30270240.L,
+ 2162160.L, 110880.L, 3960.L, 90.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ const MatrixType A8 = A6 * A2;
+ const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
+ + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (13,13)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade13(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {64764752532480000.L, 32382376266240000.L, 7771770303897600.L,
+ 1187353796428800.L, 129060195264000.L, 10559470521600.L, 670442572800.L,
+ 33522128640.L, 1323241920.L, 40840800.L, 960960.L, 16380.L, 182.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage
+ MatrixType tmp = A6 * V;
+ tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ tmp = b[12] * A6 + b[10] * A4 + b[8] * A2;
+ V.noalias() = A6 * tmp;
+ V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (17,17)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * This function activates only if your long double is double-double or quadruple.
+ */
+#if LDBL_MANT_DIG > 64
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade17(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
+ 100610229646136770560000.L, 15720348382208870400000.L,
+ 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
+ 595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
+ 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
+ 46512.L, 306.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ const MatrixType A8 = A4 * A4;
+ V = b[17] * A8 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage
+ MatrixType tmp = A8 * V;
+ tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
+ + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2;
+ V.noalias() = tmp * A8;
+ V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2
+ + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+#endif
+
+template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real>
+struct matrix_exp_computeUV
+{
+ /** \brief Compute Padé approximant to the exponential.
+ *
+ * Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Padé
+ * approximant of \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$, where \f$ M \f$
+ * denotes the matrix \c arg. 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.
+ */
+ static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings);
};
template <typename MatrixType>
-MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
- 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(M.cwiseAbs().colwise().sum().maxCoeff())
+struct matrix_exp_computeUV<MatrixType, float>
{
- /* empty body */
-}
+ template <typename ArgType>
+ static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+ {
+ using std::frexp;
+ using std::pow;
+ const float l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
+ squarings = 0;
+ if (l1norm < 4.258730016922831e-001f) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 1.880152677804762e+000f) {
+ matrix_exp_pade5(arg, U, V);
+ } else {
+ const float maxnorm = 3.925724783138660f;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<float>(squarings));
+ matrix_exp_pade7(A, U, V);
+ }
+ }
+};
template <typename MatrixType>
-template <typename ResultType>
-void MatrixExponential<MatrixType>::compute(ResultType &result)
+struct matrix_exp_computeUV<MatrixType, double>
{
-#if LDBL_MANT_DIG > 112 // rarely happens
- if(sizeof(RealScalar) > 14) {
- result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
- return;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ template <typename ArgType>
+ static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+ {
+ using std::frexp;
+ using std::pow;
+ const RealScalar l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
+ squarings = 0;
+ if (l1norm < 1.495585217958292e-002) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 2.539398330063230e-001) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 9.504178996162932e-001) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 2.097847961257068e+000) {
+ matrix_exp_pade9(arg, U, V);
+ } else {
+ const RealScalar maxnorm = 5.371920351148152;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<RealScalar>(squarings));
+ matrix_exp_pade13(A, U, V);
+ }
}
+};
+
+template <typename MatrixType>
+struct matrix_exp_computeUV<MatrixType, long double>
+{
+ template <typename ArgType>
+ static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+ {
+#if LDBL_MANT_DIG == 53 // double precision
+ matrix_exp_computeUV<MatrixType, double>::run(arg, U, V, squarings);
+
+#else
+
+ using std::frexp;
+ using std::pow;
+ const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
+ squarings = 0;
+
+#if LDBL_MANT_DIG <= 64 // extended precision
+
+ if (l1norm < 4.1968497232266989671e-003L) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 1.1848116734693823091e-001L) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 5.5170388480686700274e-001L) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 1.3759868875587845383e+000L) {
+ matrix_exp_pade9(arg, U, V);
+ } else {
+ const long double maxnorm = 4.0246098906697353063L;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
+ matrix_exp_pade13(A, U, V);
+ }
+
+#elif LDBL_MANT_DIG <= 106 // double-double
+
+ if (l1norm < 3.2787892205607026992947488108213e-005L) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 6.4467025060072760084130906076332e-003L) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 6.8988028496595374751374122881143e-002L) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 2.7339737518502231741495857201670e-001L) {
+ matrix_exp_pade9(arg, U, V);
+ } else if (l1norm < 1.3203382096514474905666448850278e+000L) {
+ matrix_exp_pade13(arg, U, V);
+ } else {
+ const long double maxnorm = 3.2579440895405400856599663723517L;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
+ matrix_exp_pade17(A, U, V);
+ }
+
+#elif LDBL_MANT_DIG <= 112 // quadruple precison
+
+ if (l1norm < 1.639394610288918690547467954466970e-005L) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 4.253237712165275566025884344433009e-003L) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 5.125804063165764409885122032933142e-002L) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 2.170000765161155195453205651889853e-001L) {
+ matrix_exp_pade9(arg, U, V);
+ } else if (l1norm < 1.125358383453143065081397882891878e+000L) {
+ matrix_exp_pade13(arg, U, V);
+ } else {
+ const long double maxnorm = 2.884233277829519311757165057717815L;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
+ matrix_exp_pade17(A, U, V);
+ }
+
+#else
+
+ // this case should be handled in compute()
+ eigen_assert(false && "Bug in MatrixExponential");
+
#endif
- computeUV(RealScalar());
- m_tmp1 = m_U + m_V; // numerator of Pade approximant
- m_tmp2 = -m_U + m_V; // denominator of Pade approximant
- result = m_tmp2.partialPivLu().solve(m_tmp1);
- for (int i=0; i<m_squarings; i++)
+#endif // LDBL_MANT_DIG
+ }
+};
+
+template<typename T> struct is_exp_known_type : false_type {};
+template<> struct is_exp_known_type<float> : true_type {};
+template<> struct is_exp_known_type<double> : true_type {};
+#if LDBL_MANT_DIG <= 112
+template<> struct is_exp_known_type<long double> : true_type {};
+#endif
+
+template <typename ArgType, typename ResultType>
+void matrix_exp_compute(const ArgType& arg, ResultType &result, true_type) // natively supported scalar type
+{
+ typedef typename ArgType::PlainObject MatrixType;
+ MatrixType U, V;
+ int squarings;
+ matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V)
+ MatrixType numer = U + V;
+ MatrixType denom = -U + V;
+ result = denom.partialPivLu().solve(numer);
+ for (int i=0; i<squarings; i++)
result *= result; // undo scaling by repeated squaring
}
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
+
+/* Computes the matrix exponential
+ *
+ * \param arg argument of matrix exponential (should be plain object)
+ * \param result variable in which result will be stored
+ */
+template <typename ArgType, typename ResultType>
+void matrix_exp_compute(const ArgType& arg, ResultType &result, false_type) // default
{
- const RealScalar 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;
+ typedef typename ArgType::PlainObject MatrixType;
+ typedef typename traits<MatrixType>::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef typename std::complex<RealScalar> ComplexScalar;
+ result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>);
}
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
-{
- const RealScalar 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 RealScalar 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 RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
- 2162160., 110880., 3960., 90., 1.};
- 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 RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
- 1187353796428800., 129060195264000., 10559470521600., 670442572800.,
- 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
- 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;
-}
-
-#if LDBL_MANT_DIG > 64
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
-{
- const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
- 100610229646136770560000.L, 15720348382208870400000.L,
- 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
- 595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
- 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
- 46512.L, 306.L, 1.L};
- MatrixType A2 = A * A;
- MatrixType A4 = A2 * A2;
- MatrixType A6 = A4 * A2;
- m_tmp1.noalias() = A4 * A4;
- m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2; // used for temporary storage
- m_tmp2.noalias() = m_tmp1 * m_V;
- 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_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
- m_V.noalias() = m_tmp1 * m_tmp2;
- m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
-}
-#endif
-
-template <typename MatrixType>
-void MatrixExponential<MatrixType>::computeUV(float)
-{
- using std::frexp;
- using std::pow;
- 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;
- frexp(m_l1norm / maxnorm, &m_squarings);
- if (m_squarings < 0) m_squarings = 0;
- MatrixType A = m_M / pow(Scalar(2), m_squarings);
- pade7(A);
- }
-}
-
-template <typename MatrixType>
-void MatrixExponential<MatrixType>::computeUV(double)
-{
- using std::frexp;
- using std::pow;
- 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;
- frexp(m_l1norm / maxnorm, &m_squarings);
- if (m_squarings < 0) m_squarings = 0;
- MatrixType A = m_M / pow(Scalar(2), m_squarings);
- pade13(A);
- }
-}
-
-template <typename MatrixType>
-void MatrixExponential<MatrixType>::computeUV(long double)
-{
- using std::frexp;
- using std::pow;
-#if LDBL_MANT_DIG == 53 // double precision
- computeUV(double());
-#elif LDBL_MANT_DIG <= 64 // extended precision
- if (m_l1norm < 4.1968497232266989671e-003L) {
- pade3(m_M);
- } else if (m_l1norm < 1.1848116734693823091e-001L) {
- pade5(m_M);
- } else if (m_l1norm < 5.5170388480686700274e-001L) {
- pade7(m_M);
- } else if (m_l1norm < 1.3759868875587845383e+000L) {
- pade9(m_M);
- } else {
- const long double maxnorm = 4.0246098906697353063L;
- frexp(m_l1norm / maxnorm, &m_squarings);
- if (m_squarings < 0) m_squarings = 0;
- MatrixType A = m_M / pow(Scalar(2), m_squarings);
- pade13(A);
- }
-#elif LDBL_MANT_DIG <= 106 // double-double
- if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
- pade3(m_M);
- } else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
- pade5(m_M);
- } else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
- pade7(m_M);
- } else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
- pade9(m_M);
- } else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
- pade13(m_M);
- } else {
- const long double maxnorm = 3.2579440895405400856599663723517L;
- frexp(m_l1norm / maxnorm, &m_squarings);
- if (m_squarings < 0) m_squarings = 0;
- MatrixType A = m_M / pow(Scalar(2), m_squarings);
- pade17(A);
- }
-#elif LDBL_MANT_DIG <= 112 // quadruple precison
- if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
- pade3(m_M);
- } else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
- pade5(m_M);
- } else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
- pade7(m_M);
- } else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
- pade9(m_M);
- } else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
- pade13(m_M);
- } else {
- const long double maxnorm = 2.884233277829519311757165057717815L;
- frexp(m_l1norm / maxnorm, &m_squarings);
- if (m_squarings < 0) m_squarings = 0;
- MatrixType A = m_M / pow(Scalar(2), m_squarings);
- pade17(A);
- }
-#else
- // this case should be handled in compute()
- eigen_assert(false && "Bug in MatrixExponential");
-#endif // LDBL_MANT_DIG
-}
+} // end namespace Eigen::internal
/** \ingroup MatrixFunctions_Module
*
@@ -391,11 +389,9 @@
*
* \tparam Derived Type of the argument to the matrix exponential.
*
- * This class holds the argument to the matrix exponential until it
- * is assigned or evaluated for some other reason (so the argument
- * should not be changed in the meantime). It is the return type of
- * MatrixBase::exp() and most of the time this is the only way it is
- * used.
+ * This class holds the argument to the matrix exponential until it is assigned or evaluated for
+ * some other reason (so the argument should not be changed in the meantime). It is the return type
+ * of MatrixBase::exp() and most of the time this is the only way it is used.
*/
template<typename Derived> struct MatrixExponentialReturnValue
: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
@@ -404,31 +400,26 @@
public:
/** \brief Constructor.
*
- * \param[in] src %Matrix (expression) forming the argument of the
- * matrix exponential.
+ * \param src %Matrix (expression) forming the argument of the matrix exponential.
*/
MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
/** \brief Compute the matrix exponential.
*
- * \param[out] result the matrix exponential of \p src in the
- * constructor.
+ * \param result the matrix exponential of \p src in the constructor.
*/
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
- const typename Derived::PlainObject srcEvaluated = m_src.eval();
- MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
- me.compute(result);
+ const typename internal::nested_eval<Derived, 10>::type tmp(m_src);
+ internal::matrix_exp_compute(tmp, result, internal::is_exp_known_type<typename Derived::Scalar>());
}
Index rows() const { return m_src.rows(); }
Index cols() const { return m_src.cols(); }
protected:
- const Derived& m_src;
- private:
- MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
+ const typename internal::ref_selector<Derived>::type m_src;
};
namespace internal {
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h b/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h
index 7d42664..3df8239 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h
@@ -1,408 +1,255 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
-// Copyright (C) 2009-2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2009-2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
-#ifndef EIGEN_MATRIX_FUNCTION
-#define EIGEN_MATRIX_FUNCTION
+#ifndef EIGEN_MATRIX_FUNCTION_H
+#define EIGEN_MATRIX_FUNCTION_H
#include "StemFunction.h"
-#include "MatrixFunctionAtomic.h"
namespace Eigen {
+namespace internal {
+
+/** \brief Maximum distance allowed between eigenvalues to be considered "close". */
+static const float matrix_function_separation = 0.1f;
+
/** \ingroup MatrixFunctions_Module
- * \brief Class for computing matrix functions.
- * \tparam MatrixType type of the argument of the matrix function,
- * expected to be an instantiation of the Matrix class template.
- * \tparam AtomicType type for computing matrix function of atomic blocks.
- * \tparam IsComplex used internally to select correct specialization.
+ * \class MatrixFunctionAtomic
+ * \brief Helper class for computing matrix functions of atomic matrices.
*
- * This class implements the Schur-Parlett algorithm for computing matrix functions. The spectrum of the
- * matrix is divided in clustered of eigenvalues that lies close together. This class delegates the
- * computation of the matrix function on every block corresponding to these clusters to an object of type
- * \p AtomicType and uses these results to compute the matrix function of the whole matrix. The class
- * \p AtomicType should have a \p compute() member function for computing the matrix function of a block.
- *
- * \sa class MatrixFunctionAtomic, class MatrixLogarithmAtomic
+ * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other.
*/
-template <typename MatrixType,
- typename AtomicType,
- int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
-class MatrixFunction
-{
- public:
-
- /** \brief Constructor.
- *
- * \param[in] A argument of matrix function, should be a square matrix.
- * \param[in] atomic class for computing matrix function of atomic blocks.
- *
- * The class stores references to \p A and \p atomic, so they should not be
- * changed (or destroyed) before compute() is called.
- */
- MatrixFunction(const MatrixType& A, AtomicType& atomic);
-
- /** \brief Compute the matrix function.
- *
- * \param[out] result the function \p f applied to \p A, as
- * specified in the constructor.
- *
- * See MatrixBase::matrixFunction() for details on how this computation
- * is implemented.
- */
- template <typename ResultType>
- void compute(ResultType &result);
-};
-
-
-/** \internal \ingroup MatrixFunctions_Module
- * \brief Partial specialization of MatrixFunction for real matrices
- */
-template <typename MatrixType, typename AtomicType>
-class MatrixFunction<MatrixType, AtomicType, 0>
-{
- private:
-
- typedef internal::traits<MatrixType> Traits;
- typedef typename Traits::Scalar Scalar;
- static const int Rows = Traits::RowsAtCompileTime;
- static const int Cols = Traits::ColsAtCompileTime;
- static const int Options = MatrixType::Options;
- static const int MaxRows = Traits::MaxRowsAtCompileTime;
- static const int MaxCols = Traits::MaxColsAtCompileTime;
-
- typedef std::complex<Scalar> ComplexScalar;
- typedef Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols> ComplexMatrix;
-
- public:
-
- /** \brief Constructor.
- *
- * \param[in] A argument of matrix function, should be a square matrix.
- * \param[in] atomic class for computing matrix function of atomic blocks.
- */
- MatrixFunction(const MatrixType& A, AtomicType& atomic) : m_A(A), m_atomic(atomic) { }
-
- /** \brief Compute the matrix function.
- *
- * \param[out] result the function \p f applied to \p A, as
- * specified in the constructor.
- *
- * This function converts the real matrix \c A to a complex matrix,
- * uses MatrixFunction<MatrixType,1> and then converts the result back to
- * a real matrix.
- */
- template <typename ResultType>
- void compute(ResultType& result)
- {
- ComplexMatrix CA = m_A.template cast<ComplexScalar>();
- ComplexMatrix Cresult;
- MatrixFunction<ComplexMatrix, AtomicType> mf(CA, m_atomic);
- mf.compute(Cresult);
- result = Cresult.real();
- }
-
- private:
- typename internal::nested<MatrixType>::type m_A; /**< \brief Reference to argument of matrix function. */
- AtomicType& m_atomic; /**< \brief Class for computing matrix function of atomic blocks. */
-
- MatrixFunction& operator=(const MatrixFunction&);
-};
-
-
-/** \internal \ingroup MatrixFunctions_Module
- * \brief Partial specialization of MatrixFunction for complex matrices
- */
-template <typename MatrixType, typename AtomicType>
-class MatrixFunction<MatrixType, AtomicType, 1>
+template <typename MatrixType>
+class MatrixFunctionAtomic
{
- private:
+ public:
- typedef internal::traits<MatrixType> Traits;
typedef typename MatrixType::Scalar Scalar;
- typedef typename MatrixType::Index Index;
- static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
- static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
- static const int Options = MatrixType::Options;
- typedef typename NumTraits<Scalar>::Real RealScalar;
- typedef Matrix<Scalar, Traits::RowsAtCompileTime, 1> VectorType;
- typedef Matrix<Index, Traits::RowsAtCompileTime, 1> IntVectorType;
- typedef Matrix<Index, Dynamic, 1> DynamicIntVectorType;
- typedef std::list<Scalar> Cluster;
- typedef std::list<Cluster> ListOfClusters;
- typedef Matrix<Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
+ typedef typename stem_function<Scalar>::type StemFunction;
- public:
+ /** \brief Constructor
+ * \param[in] f matrix function to compute.
+ */
+ MatrixFunctionAtomic(StemFunction f) : m_f(f) { }
- MatrixFunction(const MatrixType& A, AtomicType& atomic);
- template <typename ResultType> void compute(ResultType& result);
+ /** \brief Compute matrix function of atomic matrix
+ * \param[in] A argument of matrix function, should be upper triangular and atomic
+ * \returns f(A), the matrix function evaluated at the given matrix
+ */
+ MatrixType compute(const MatrixType& A);
private:
-
- void computeSchurDecomposition();
- void partitionEigenvalues();
- typename ListOfClusters::iterator findCluster(Scalar key);
- void computeClusterSize();
- void computeBlockStart();
- void constructPermutation();
- void permuteSchur();
- void swapEntriesInSchur(Index index);
- void computeBlockAtomic();
- Block<MatrixType> block(MatrixType& A, Index i, Index j);
- void computeOffDiagonal();
- DynMatrixType solveTriangularSylvester(const DynMatrixType& A, const DynMatrixType& B, const DynMatrixType& C);
-
- typename internal::nested<MatrixType>::type m_A; /**< \brief Reference to argument of matrix function. */
- AtomicType& m_atomic; /**< \brief Class for computing matrix function of atomic blocks. */
- MatrixType m_T; /**< \brief Triangular part of Schur decomposition */
- MatrixType m_U; /**< \brief Unitary part of Schur decomposition */
- MatrixType m_fT; /**< \brief %Matrix function applied to #m_T */
- ListOfClusters m_clusters; /**< \brief Partition of eigenvalues into clusters of ei'vals "close" to each other */
- DynamicIntVectorType m_eivalToCluster; /**< \brief m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster */
- DynamicIntVectorType m_clusterSize; /**< \brief Number of eigenvalues in each clusters */
- DynamicIntVectorType m_blockStart; /**< \brief Row index at which block corresponding to i-th cluster starts */
- IntVectorType m_permutation; /**< \brief Permutation which groups ei'vals in the same cluster together */
-
- /** \brief Maximum distance allowed between eigenvalues to be considered "close".
- *
- * This is morally a \c static \c const \c Scalar, but only
- * integers can be static constant class members in C++. The
- * separation constant is set to 0.1, a value taken from the
- * paper by Davies and Higham. */
- static const RealScalar separation() { return static_cast<RealScalar>(0.1); }
-
- MatrixFunction& operator=(const MatrixFunction&);
+ StemFunction* m_f;
};
-/** \brief Constructor.
- *
- * \param[in] A argument of matrix function, should be a square matrix.
- * \param[in] atomic class for computing matrix function of atomic blocks.
- */
-template <typename MatrixType, typename AtomicType>
-MatrixFunction<MatrixType,AtomicType,1>::MatrixFunction(const MatrixType& A, AtomicType& atomic)
- : m_A(A), m_atomic(atomic)
+template <typename MatrixType>
+typename NumTraits<typename MatrixType::Scalar>::Real matrix_function_compute_mu(const MatrixType& A)
{
- /* empty body */
+ typedef typename plain_col_type<MatrixType>::type VectorType;
+ typename MatrixType::Index rows = A.rows();
+ const MatrixType N = MatrixType::Identity(rows, rows) - A;
+ VectorType e = VectorType::Ones(rows);
+ N.template triangularView<Upper>().solveInPlace(e);
+ return e.cwiseAbs().maxCoeff();
}
-/** \brief Compute the matrix function.
- *
- * \param[out] result the function \p f applied to \p A, as
- * specified in the constructor.
+template <typename MatrixType>
+MatrixType MatrixFunctionAtomic<MatrixType>::compute(const MatrixType& A)
+{
+ // TODO: Use that A is upper triangular
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef typename MatrixType::Index Index;
+ Index rows = A.rows();
+ Scalar avgEival = A.trace() / Scalar(RealScalar(rows));
+ MatrixType Ashifted = A - avgEival * MatrixType::Identity(rows, rows);
+ RealScalar mu = matrix_function_compute_mu(Ashifted);
+ MatrixType F = m_f(avgEival, 0) * MatrixType::Identity(rows, rows);
+ MatrixType P = Ashifted;
+ MatrixType Fincr;
+ for (Index s = 1; s < 1.1 * rows + 10; s++) { // upper limit is fairly arbitrary
+ Fincr = m_f(avgEival, static_cast<int>(s)) * P;
+ F += Fincr;
+ P = Scalar(RealScalar(1.0/(s + 1))) * P * Ashifted;
+
+ // test whether Taylor series converged
+ const RealScalar F_norm = F.cwiseAbs().rowwise().sum().maxCoeff();
+ const RealScalar Fincr_norm = Fincr.cwiseAbs().rowwise().sum().maxCoeff();
+ if (Fincr_norm < NumTraits<Scalar>::epsilon() * F_norm) {
+ RealScalar delta = 0;
+ RealScalar rfactorial = 1;
+ for (Index r = 0; r < rows; r++) {
+ RealScalar mx = 0;
+ for (Index i = 0; i < rows; i++)
+ mx = (std::max)(mx, std::abs(m_f(Ashifted(i, i) + avgEival, static_cast<int>(s+r))));
+ if (r != 0)
+ rfactorial *= RealScalar(r);
+ delta = (std::max)(delta, mx / rfactorial);
+ }
+ const RealScalar P_norm = P.cwiseAbs().rowwise().sum().maxCoeff();
+ if (mu * delta * P_norm < NumTraits<Scalar>::epsilon() * F_norm) // series converged
+ break;
+ }
+ }
+ return F;
+}
+
+/** \brief Find cluster in \p clusters containing some value
+ * \param[in] key Value to find
+ * \returns Iterator to cluster containing \p key, or \c clusters.end() if no cluster in \p m_clusters
+ * contains \p key.
*/
-template <typename MatrixType, typename AtomicType>
-template <typename ResultType>
-void MatrixFunction<MatrixType,AtomicType,1>::compute(ResultType& result)
+template <typename Index, typename ListOfClusters>
+typename ListOfClusters::iterator matrix_function_find_cluster(Index key, ListOfClusters& clusters)
{
- computeSchurDecomposition();
- partitionEigenvalues();
- computeClusterSize();
- computeBlockStart();
- constructPermutation();
- permuteSchur();
- computeBlockAtomic();
- computeOffDiagonal();
- result = m_U * (m_fT.template triangularView<Upper>() * m_U.adjoint());
-}
-
-/** \brief Store the Schur decomposition of #m_A in #m_T and #m_U */
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::computeSchurDecomposition()
-{
- const ComplexSchur<MatrixType> schurOfA(m_A);
- m_T = schurOfA.matrixT();
- m_U = schurOfA.matrixU();
+ typename std::list<Index>::iterator j;
+ for (typename ListOfClusters::iterator i = clusters.begin(); i != clusters.end(); ++i) {
+ j = std::find(i->begin(), i->end(), key);
+ if (j != i->end())
+ return i;
+ }
+ return clusters.end();
}
/** \brief Partition eigenvalues in clusters of ei'vals close to each other
*
- * This function computes #m_clusters. This is a partition of the
- * eigenvalues of #m_T in clusters, such that
- * # Any eigenvalue in a certain cluster is at most separation() away
- * from another eigenvalue in the same cluster.
- * # The distance between two eigenvalues in different clusters is
- * more than separation().
- * The implementation follows Algorithm 4.1 in the paper of Davies
- * and Higham.
+ * \param[in] eivals Eigenvalues
+ * \param[out] clusters Resulting partition of eigenvalues
+ *
+ * The partition satisfies the following two properties:
+ * # Any eigenvalue in a certain cluster is at most matrix_function_separation() away from another eigenvalue
+ * in the same cluster.
+ * # The distance between two eigenvalues in different clusters is more than matrix_function_separation().
+ * The implementation follows Algorithm 4.1 in the paper of Davies and Higham.
*/
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::partitionEigenvalues()
+template <typename EivalsType, typename Cluster>
+void matrix_function_partition_eigenvalues(const EivalsType& eivals, std::list<Cluster>& clusters)
{
- using std::abs;
- const Index rows = m_T.rows();
- VectorType diag = m_T.diagonal(); // contains eigenvalues of A
-
- for (Index i=0; i<rows; ++i) {
- // Find set containing diag(i), adding a new set if necessary
- typename ListOfClusters::iterator qi = findCluster(diag(i));
- if (qi == m_clusters.end()) {
+ typedef typename EivalsType::Index Index;
+ typedef typename EivalsType::RealScalar RealScalar;
+ for (Index i=0; i<eivals.rows(); ++i) {
+ // Find cluster containing i-th ei'val, adding a new cluster if necessary
+ typename std::list<Cluster>::iterator qi = matrix_function_find_cluster(i, clusters);
+ if (qi == clusters.end()) {
Cluster l;
- l.push_back(diag(i));
- m_clusters.push_back(l);
- qi = m_clusters.end();
+ l.push_back(i);
+ clusters.push_back(l);
+ qi = clusters.end();
--qi;
}
// Look for other element to add to the set
- for (Index j=i+1; j<rows; ++j) {
- if (abs(diag(j) - diag(i)) <= separation() && std::find(qi->begin(), qi->end(), diag(j)) == qi->end()) {
- typename ListOfClusters::iterator qj = findCluster(diag(j));
- if (qj == m_clusters.end()) {
- qi->push_back(diag(j));
+ for (Index j=i+1; j<eivals.rows(); ++j) {
+ if (abs(eivals(j) - eivals(i)) <= RealScalar(matrix_function_separation)
+ && std::find(qi->begin(), qi->end(), j) == qi->end()) {
+ typename std::list<Cluster>::iterator qj = matrix_function_find_cluster(j, clusters);
+ if (qj == clusters.end()) {
+ qi->push_back(j);
} else {
qi->insert(qi->end(), qj->begin(), qj->end());
- m_clusters.erase(qj);
+ clusters.erase(qj);
}
}
}
}
}
-/** \brief Find cluster in #m_clusters containing some value
- * \param[in] key Value to find
- * \returns Iterator to cluster containing \c key, or
- * \c m_clusters.end() if no cluster in m_clusters contains \c key.
- */
-template <typename MatrixType, typename AtomicType>
-typename MatrixFunction<MatrixType,AtomicType,1>::ListOfClusters::iterator MatrixFunction<MatrixType,AtomicType,1>::findCluster(Scalar key)
+/** \brief Compute size of each cluster given a partitioning */
+template <typename ListOfClusters, typename Index>
+void matrix_function_compute_cluster_size(const ListOfClusters& clusters, Matrix<Index, Dynamic, 1>& clusterSize)
{
- typename Cluster::iterator j;
- for (typename ListOfClusters::iterator i = m_clusters.begin(); i != m_clusters.end(); ++i) {
- j = std::find(i->begin(), i->end(), key);
- if (j != i->end())
- return i;
+ const Index numClusters = static_cast<Index>(clusters.size());
+ clusterSize.setZero(numClusters);
+ Index clusterIndex = 0;
+ for (typename ListOfClusters::const_iterator cluster = clusters.begin(); cluster != clusters.end(); ++cluster) {
+ clusterSize[clusterIndex] = cluster->size();
+ ++clusterIndex;
}
- return m_clusters.end();
}
-/** \brief Compute #m_clusterSize and #m_eivalToCluster using #m_clusters */
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::computeClusterSize()
+/** \brief Compute start of each block using clusterSize */
+template <typename VectorType>
+void matrix_function_compute_block_start(const VectorType& clusterSize, VectorType& blockStart)
{
- const Index rows = m_T.rows();
- VectorType diag = m_T.diagonal();
- const Index numClusters = static_cast<Index>(m_clusters.size());
+ blockStart.resize(clusterSize.rows());
+ blockStart(0) = 0;
+ for (typename VectorType::Index i = 1; i < clusterSize.rows(); i++) {
+ blockStart(i) = blockStart(i-1) + clusterSize(i-1);
+ }
+}
- m_clusterSize.setZero(numClusters);
- m_eivalToCluster.resize(rows);
+/** \brief Compute mapping of eigenvalue indices to cluster indices */
+template <typename EivalsType, typename ListOfClusters, typename VectorType>
+void matrix_function_compute_map(const EivalsType& eivals, const ListOfClusters& clusters, VectorType& eivalToCluster)
+{
+ typedef typename EivalsType::Index Index;
+ eivalToCluster.resize(eivals.rows());
Index clusterIndex = 0;
- for (typename ListOfClusters::const_iterator cluster = m_clusters.begin(); cluster != m_clusters.end(); ++cluster) {
- for (Index i = 0; i < diag.rows(); ++i) {
- if (std::find(cluster->begin(), cluster->end(), diag(i)) != cluster->end()) {
- ++m_clusterSize[clusterIndex];
- m_eivalToCluster[i] = clusterIndex;
+ for (typename ListOfClusters::const_iterator cluster = clusters.begin(); cluster != clusters.end(); ++cluster) {
+ for (Index i = 0; i < eivals.rows(); ++i) {
+ if (std::find(cluster->begin(), cluster->end(), i) != cluster->end()) {
+ eivalToCluster[i] = clusterIndex;
}
}
++clusterIndex;
}
}
-/** \brief Compute #m_blockStart using #m_clusterSize */
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::computeBlockStart()
+/** \brief Compute permutation which groups ei'vals in same cluster together */
+template <typename DynVectorType, typename VectorType>
+void matrix_function_compute_permutation(const DynVectorType& blockStart, const DynVectorType& eivalToCluster, VectorType& permutation)
{
- m_blockStart.resize(m_clusterSize.rows());
- m_blockStart(0) = 0;
- for (Index i = 1; i < m_clusterSize.rows(); i++) {
- m_blockStart(i) = m_blockStart(i-1) + m_clusterSize(i-1);
- }
-}
-
-/** \brief Compute #m_permutation using #m_eivalToCluster and #m_blockStart */
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::constructPermutation()
-{
- DynamicIntVectorType indexNextEntry = m_blockStart;
- m_permutation.resize(m_T.rows());
- for (Index i = 0; i < m_T.rows(); i++) {
- Index cluster = m_eivalToCluster[i];
- m_permutation[i] = indexNextEntry[cluster];
+ typedef typename VectorType::Index Index;
+ DynVectorType indexNextEntry = blockStart;
+ permutation.resize(eivalToCluster.rows());
+ for (Index i = 0; i < eivalToCluster.rows(); i++) {
+ Index cluster = eivalToCluster[i];
+ permutation[i] = indexNextEntry[cluster];
++indexNextEntry[cluster];
}
}
-/** \brief Permute Schur decomposition in #m_U and #m_T according to #m_permutation */
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::permuteSchur()
+/** \brief Permute Schur decomposition in U and T according to permutation */
+template <typename VectorType, typename MatrixType>
+void matrix_function_permute_schur(VectorType& permutation, MatrixType& U, MatrixType& T)
{
- IntVectorType p = m_permutation;
- for (Index i = 0; i < p.rows() - 1; i++) {
+ typedef typename VectorType::Index Index;
+ for (Index i = 0; i < permutation.rows() - 1; i++) {
Index j;
- for (j = i; j < p.rows(); j++) {
- if (p(j) == i) break;
+ for (j = i; j < permutation.rows(); j++) {
+ if (permutation(j) == i) break;
}
- eigen_assert(p(j) == i);
+ eigen_assert(permutation(j) == i);
for (Index k = j-1; k >= i; k--) {
- swapEntriesInSchur(k);
- std::swap(p.coeffRef(k), p.coeffRef(k+1));
+ JacobiRotation<typename MatrixType::Scalar> rotation;
+ rotation.makeGivens(T(k, k+1), T(k+1, k+1) - T(k, k));
+ T.applyOnTheLeft(k, k+1, rotation.adjoint());
+ T.applyOnTheRight(k, k+1, rotation);
+ U.applyOnTheRight(k, k+1, rotation);
+ std::swap(permutation.coeffRef(k), permutation.coeffRef(k+1));
}
}
}
-/** \brief Swap rows \a index and \a index+1 in Schur decomposition in #m_U and #m_T */
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::swapEntriesInSchur(Index index)
-{
- JacobiRotation<Scalar> rotation;
- rotation.makeGivens(m_T(index, index+1), m_T(index+1, index+1) - m_T(index, index));
- m_T.applyOnTheLeft(index, index+1, rotation.adjoint());
- m_T.applyOnTheRight(index, index+1, rotation);
- m_U.applyOnTheRight(index, index+1, rotation);
-}
-
-/** \brief Compute block diagonal part of #m_fT.
+/** \brief Compute block diagonal part of matrix function.
*
- * This routine computes the matrix function applied to the block diagonal part of #m_T, with the blocking
- * given by #m_blockStart. The matrix function of each diagonal block is computed by #m_atomic. The
- * off-diagonal parts of #m_fT are set to zero.
+ * This routine computes the matrix function applied to the block diagonal part of \p T (which should be
+ * upper triangular), with the blocking given by \p blockStart and \p clusterSize. The matrix function of
+ * each diagonal block is computed by \p atomic. The off-diagonal parts of \p fT are set to zero.
*/
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::computeBlockAtomic()
+template <typename MatrixType, typename AtomicType, typename VectorType>
+void matrix_function_compute_block_atomic(const MatrixType& T, AtomicType& atomic, const VectorType& blockStart, const VectorType& clusterSize, MatrixType& fT)
{
- m_fT.resize(m_T.rows(), m_T.cols());
- m_fT.setZero();
- for (Index i = 0; i < m_clusterSize.rows(); ++i) {
- block(m_fT, i, i) = m_atomic.compute(block(m_T, i, i));
- }
-}
-
-/** \brief Return block of matrix according to blocking given by #m_blockStart */
-template <typename MatrixType, typename AtomicType>
-Block<MatrixType> MatrixFunction<MatrixType,AtomicType,1>::block(MatrixType& A, Index i, Index j)
-{
- return A.block(m_blockStart(i), m_blockStart(j), m_clusterSize(i), m_clusterSize(j));
-}
-
-/** \brief Compute part of #m_fT above block diagonal.
- *
- * This routine assumes that the block diagonal part of #m_fT (which
- * equals the matrix function applied to #m_T) has already been computed and computes
- * the part above the block diagonal. The part below the diagonal is
- * zero, because #m_T is upper triangular.
- */
-template <typename MatrixType, typename AtomicType>
-void MatrixFunction<MatrixType,AtomicType,1>::computeOffDiagonal()
-{
- for (Index diagIndex = 1; diagIndex < m_clusterSize.rows(); diagIndex++) {
- for (Index blockIndex = 0; blockIndex < m_clusterSize.rows() - diagIndex; blockIndex++) {
- // compute (blockIndex, blockIndex+diagIndex) block
- DynMatrixType A = block(m_T, blockIndex, blockIndex);
- DynMatrixType B = -block(m_T, blockIndex+diagIndex, blockIndex+diagIndex);
- DynMatrixType C = block(m_fT, blockIndex, blockIndex) * block(m_T, blockIndex, blockIndex+diagIndex);
- C -= block(m_T, blockIndex, blockIndex+diagIndex) * block(m_fT, blockIndex+diagIndex, blockIndex+diagIndex);
- for (Index k = blockIndex + 1; k < blockIndex + diagIndex; k++) {
- C += block(m_fT, blockIndex, k) * block(m_T, k, blockIndex+diagIndex);
- C -= block(m_T, blockIndex, k) * block(m_fT, k, blockIndex+diagIndex);
- }
- block(m_fT, blockIndex, blockIndex+diagIndex) = solveTriangularSylvester(A, B, C);
- }
+ fT.setZero(T.rows(), T.cols());
+ for (typename VectorType::Index i = 0; i < clusterSize.rows(); ++i) {
+ fT.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i))
+ = atomic.compute(T.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i)));
}
}
@@ -414,8 +261,8 @@
*
* \returns the solution X.
*
- * If A is m-by-m and B is n-by-n, then both C and X are m-by-n.
- * The (i,j)-th component of the Sylvester equation is
+ * If A is m-by-m and B is n-by-n, then both C and X are m-by-n. The (i,j)-th component of the Sylvester
+ * equation is
* \f[
* \sum_{k=i}^m A_{ik} X_{kj} + \sum_{k=1}^j X_{ik} B_{kj} = C_{ij}.
* \f]
@@ -424,16 +271,12 @@
* X_{ij} = \frac{1}{A_{ii} + B_{jj}} \Bigl( C_{ij}
* - \sum_{k=i+1}^m A_{ik} X_{kj} - \sum_{k=1}^{j-1} X_{ik} B_{kj} \Bigr).
* \f]
- * It is assumed that A and B are such that the numerator is never
- * zero (otherwise the Sylvester equation does not have a unique
- * solution). In that case, these equations can be evaluated in the
- * order \f$ i=m,\ldots,1 \f$ and \f$ j=1,\ldots,n \f$.
+ * It is assumed that A and B are such that the numerator is never zero (otherwise the Sylvester equation
+ * does not have a unique solution). In that case, these equations can be evaluated in the order
+ * \f$ i=m,\ldots,1 \f$ and \f$ j=1,\ldots,n \f$.
*/
-template <typename MatrixType, typename AtomicType>
-typename MatrixFunction<MatrixType,AtomicType,1>::DynMatrixType MatrixFunction<MatrixType,AtomicType,1>::solveTriangularSylvester(
- const DynMatrixType& A,
- const DynMatrixType& B,
- const DynMatrixType& C)
+template <typename MatrixType>
+MatrixType matrix_function_solve_triangular_sylvester(const MatrixType& A, const MatrixType& B, const MatrixType& C)
{
eigen_assert(A.rows() == A.cols());
eigen_assert(A.isUpperTriangular());
@@ -442,9 +285,12 @@
eigen_assert(C.rows() == A.rows());
eigen_assert(C.cols() == B.rows());
+ typedef typename MatrixType::Index Index;
+ typedef typename MatrixType::Scalar Scalar;
+
Index m = A.rows();
Index n = B.rows();
- DynMatrixType X(m, n);
+ MatrixType X(m, n);
for (Index i = m - 1; i >= 0; --i) {
for (Index j = 0; j < n; ++j) {
@@ -473,66 +319,209 @@
return X;
}
+/** \brief Compute part of matrix function above block diagonal.
+ *
+ * This routine completes the computation of \p fT, denoting a matrix function applied to the triangular
+ * matrix \p T. It assumes that the block diagonal part of \p fT has already been computed. The part below
+ * the diagonal is zero, because \p T is upper triangular.
+ */
+template <typename MatrixType, typename VectorType>
+void matrix_function_compute_above_diagonal(const MatrixType& T, const VectorType& blockStart, const VectorType& clusterSize, MatrixType& fT)
+{
+ typedef internal::traits<MatrixType> Traits;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename MatrixType::Index Index;
+ static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
+ static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
+ static const int Options = MatrixType::Options;
+ typedef Matrix<Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
+
+ for (Index k = 1; k < clusterSize.rows(); k++) {
+ for (Index i = 0; i < clusterSize.rows() - k; i++) {
+ // compute (i, i+k) block
+ DynMatrixType A = T.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i));
+ DynMatrixType B = -T.block(blockStart(i+k), blockStart(i+k), clusterSize(i+k), clusterSize(i+k));
+ DynMatrixType C = fT.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i))
+ * T.block(blockStart(i), blockStart(i+k), clusterSize(i), clusterSize(i+k));
+ C -= T.block(blockStart(i), blockStart(i+k), clusterSize(i), clusterSize(i+k))
+ * fT.block(blockStart(i+k), blockStart(i+k), clusterSize(i+k), clusterSize(i+k));
+ for (Index m = i + 1; m < i + k; m++) {
+ C += fT.block(blockStart(i), blockStart(m), clusterSize(i), clusterSize(m))
+ * T.block(blockStart(m), blockStart(i+k), clusterSize(m), clusterSize(i+k));
+ C -= T.block(blockStart(i), blockStart(m), clusterSize(i), clusterSize(m))
+ * fT.block(blockStart(m), blockStart(i+k), clusterSize(m), clusterSize(i+k));
+ }
+ fT.block(blockStart(i), blockStart(i+k), clusterSize(i), clusterSize(i+k))
+ = matrix_function_solve_triangular_sylvester(A, B, C);
+ }
+ }
+}
+
+/** \ingroup MatrixFunctions_Module
+ * \brief Class for computing matrix functions.
+ * \tparam MatrixType type of the argument of the matrix function,
+ * expected to be an instantiation of the Matrix class template.
+ * \tparam AtomicType type for computing matrix function of atomic blocks.
+ * \tparam IsComplex used internally to select correct specialization.
+ *
+ * This class implements the Schur-Parlett algorithm for computing matrix functions. The spectrum of the
+ * matrix is divided in clustered of eigenvalues that lies close together. This class delegates the
+ * computation of the matrix function on every block corresponding to these clusters to an object of type
+ * \p AtomicType and uses these results to compute the matrix function of the whole matrix. The class
+ * \p AtomicType should have a \p compute() member function for computing the matrix function of a block.
+ *
+ * \sa class MatrixFunctionAtomic, class MatrixLogarithmAtomic
+ */
+template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
+struct matrix_function_compute
+{
+ /** \brief Compute the matrix function.
+ *
+ * \param[in] A argument of matrix function, should be a square matrix.
+ * \param[in] atomic class for computing matrix function of atomic blocks.
+ * \param[out] result the function \p f applied to \p A, as
+ * specified in the constructor.
+ *
+ * See MatrixBase::matrixFunction() for details on how this computation
+ * is implemented.
+ */
+ template <typename AtomicType, typename ResultType>
+ static void run(const MatrixType& A, AtomicType& atomic, ResultType &result);
+};
+
+/** \internal \ingroup MatrixFunctions_Module
+ * \brief Partial specialization of MatrixFunction for real matrices
+ *
+ * This converts the real matrix to a complex matrix, compute the matrix function of that matrix, and then
+ * converts the result back to a real matrix.
+ */
+template <typename MatrixType>
+struct matrix_function_compute<MatrixType, 0>
+{
+ template <typename MatA, typename AtomicType, typename ResultType>
+ static void run(const MatA& A, AtomicType& atomic, ResultType &result)
+ {
+ typedef internal::traits<MatrixType> Traits;
+ typedef typename Traits::Scalar Scalar;
+ static const int Rows = Traits::RowsAtCompileTime, Cols = Traits::ColsAtCompileTime;
+ static const int MaxRows = Traits::MaxRowsAtCompileTime, MaxCols = Traits::MaxColsAtCompileTime;
+
+ typedef std::complex<Scalar> ComplexScalar;
+ typedef Matrix<ComplexScalar, Rows, Cols, 0, MaxRows, MaxCols> ComplexMatrix;
+
+ ComplexMatrix CA = A.template cast<ComplexScalar>();
+ ComplexMatrix Cresult;
+ matrix_function_compute<ComplexMatrix>::run(CA, atomic, Cresult);
+ result = Cresult.real();
+ }
+};
+
+/** \internal \ingroup MatrixFunctions_Module
+ * \brief Partial specialization of MatrixFunction for complex matrices
+ */
+template <typename MatrixType>
+struct matrix_function_compute<MatrixType, 1>
+{
+ template <typename MatA, typename AtomicType, typename ResultType>
+ static void run(const MatA& A, AtomicType& atomic, ResultType &result)
+ {
+ typedef internal::traits<MatrixType> Traits;
+
+ // compute Schur decomposition of A
+ const ComplexSchur<MatrixType> schurOfA(A);
+ MatrixType T = schurOfA.matrixT();
+ MatrixType U = schurOfA.matrixU();
+
+ // partition eigenvalues into clusters of ei'vals "close" to each other
+ std::list<std::list<Index> > clusters;
+ matrix_function_partition_eigenvalues(T.diagonal(), clusters);
+
+ // compute size of each cluster
+ Matrix<Index, Dynamic, 1> clusterSize;
+ matrix_function_compute_cluster_size(clusters, clusterSize);
+
+ // blockStart[i] is row index at which block corresponding to i-th cluster starts
+ Matrix<Index, Dynamic, 1> blockStart;
+ matrix_function_compute_block_start(clusterSize, blockStart);
+
+ // compute map so that eivalToCluster[i] = j means that i-th ei'val is in j-th cluster
+ Matrix<Index, Dynamic, 1> eivalToCluster;
+ matrix_function_compute_map(T.diagonal(), clusters, eivalToCluster);
+
+ // compute permutation which groups ei'vals in same cluster together
+ Matrix<Index, Traits::RowsAtCompileTime, 1> permutation;
+ matrix_function_compute_permutation(blockStart, eivalToCluster, permutation);
+
+ // permute Schur decomposition
+ matrix_function_permute_schur(permutation, U, T);
+
+ // compute result
+ MatrixType fT; // matrix function applied to T
+ matrix_function_compute_block_atomic(T, atomic, blockStart, clusterSize, fT);
+ matrix_function_compute_above_diagonal(T, blockStart, clusterSize, fT);
+ result = U * (fT.template triangularView<Upper>() * U.adjoint());
+ }
+};
+
+} // end of namespace internal
+
/** \ingroup MatrixFunctions_Module
*
* \brief Proxy for the matrix function of some matrix (expression).
*
* \tparam Derived Type of the argument to the matrix function.
*
- * This class holds the argument to the matrix function until it is
- * assigned or evaluated for some other reason (so the argument
- * should not be changed in the meantime). It is the return type of
- * matrixBase::matrixFunction() and related functions and most of the
- * time this is the only way it is used.
+ * This class holds the argument to the matrix function until it is assigned or evaluated for some other
+ * reason (so the argument should not be changed in the meantime). It is the return type of
+ * matrixBase::matrixFunction() and related functions and most of the time this is the only way it is used.
*/
template<typename Derived> class MatrixFunctionReturnValue
: public ReturnByValue<MatrixFunctionReturnValue<Derived> >
{
public:
-
typedef typename Derived::Scalar Scalar;
typedef typename Derived::Index Index;
typedef typename internal::stem_function<Scalar>::type StemFunction;
- /** \brief Constructor.
+ protected:
+ typedef typename internal::ref_selector<Derived>::type DerivedNested;
+
+ public:
+
+ /** \brief Constructor.
*
- * \param[in] A %Matrix (expression) forming the argument of the
- * matrix function.
+ * \param[in] A %Matrix (expression) forming the argument of the matrix function.
* \param[in] f Stem function for matrix function under consideration.
*/
MatrixFunctionReturnValue(const Derived& A, StemFunction f) : m_A(A), m_f(f) { }
/** \brief Compute the matrix function.
*
- * \param[out] result \p f applied to \p A, where \p f and \p A
- * are as in the constructor.
+ * \param[out] result \p f applied to \p A, where \p f and \p A are as in the constructor.
*/
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
- typedef typename Derived::PlainObject PlainObject;
- typedef internal::traits<PlainObject> Traits;
+ typedef typename internal::nested_eval<Derived, 10>::type NestedEvalType;
+ typedef typename internal::remove_all<NestedEvalType>::type NestedEvalTypeClean;
+ typedef internal::traits<NestedEvalTypeClean> Traits;
static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
- static const int Options = PlainObject::Options;
typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
- typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
- typedef MatrixFunctionAtomic<DynMatrixType> AtomicType;
+ typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
+
+ typedef internal::MatrixFunctionAtomic<DynMatrixType> AtomicType;
AtomicType atomic(m_f);
- const PlainObject Aevaluated = m_A.eval();
- MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
- mf.compute(result);
+ internal::matrix_function_compute<typename NestedEvalTypeClean::PlainObject>::run(m_A, atomic, result);
}
Index rows() const { return m_A.rows(); }
Index cols() const { return m_A.cols(); }
private:
- typename internal::nested<Derived>::type m_A;
+ const DerivedNested m_A;
StemFunction *m_f;
-
- MatrixFunctionReturnValue& operator=(const MatrixFunctionReturnValue&);
};
namespace internal {
@@ -559,7 +548,7 @@
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
- return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sin);
+ return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_sin<ComplexScalar>);
}
template <typename Derived>
@@ -567,7 +556,7 @@
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
- return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::cos);
+ return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_cos<ComplexScalar>);
}
template <typename Derived>
@@ -575,7 +564,7 @@
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
- return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sinh);
+ return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_sinh<ComplexScalar>);
}
template <typename Derived>
@@ -583,9 +572,9 @@
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
- return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::cosh);
+ return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_cosh<ComplexScalar>);
}
} // end namespace Eigen
-#endif // EIGEN_MATRIX_FUNCTION
+#endif // EIGEN_MATRIX_FUNCTION_H
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixFunctionAtomic.h b/unsupported/Eigen/src/MatrixFunctions/MatrixFunctionAtomic.h
deleted file mode 100644
index efe332c..0000000
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixFunctionAtomic.h
+++ /dev/null
@@ -1,131 +0,0 @@
-// This file is part of Eigen, a lightweight C++ template library
-// for linear algebra.
-//
-// Copyright (C) 2009 Jitse Niesen <jitse@maths.leeds.ac.uk>
-//
-// This Source Code Form is subject to the terms of the Mozilla
-// Public License v. 2.0. If a copy of the MPL was not distributed
-// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
-
-#ifndef EIGEN_MATRIX_FUNCTION_ATOMIC
-#define EIGEN_MATRIX_FUNCTION_ATOMIC
-
-namespace Eigen {
-
-/** \ingroup MatrixFunctions_Module
- * \class MatrixFunctionAtomic
- * \brief Helper class for computing matrix functions of atomic matrices.
- *
- * \internal
- * Here, an atomic matrix is a triangular matrix whose diagonal
- * entries are close to each other.
- */
-template <typename MatrixType>
-class MatrixFunctionAtomic
-{
- public:
-
- typedef typename MatrixType::Scalar Scalar;
- typedef typename MatrixType::Index Index;
- typedef typename NumTraits<Scalar>::Real RealScalar;
- typedef typename internal::stem_function<Scalar>::type StemFunction;
- typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
-
- /** \brief Constructor
- * \param[in] f matrix function to compute.
- */
- MatrixFunctionAtomic(StemFunction f) : m_f(f) { }
-
- /** \brief Compute matrix function of atomic matrix
- * \param[in] A argument of matrix function, should be upper triangular and atomic
- * \returns f(A), the matrix function evaluated at the given matrix
- */
- MatrixType compute(const MatrixType& A);
-
- private:
-
- // Prevent copying
- MatrixFunctionAtomic(const MatrixFunctionAtomic&);
- MatrixFunctionAtomic& operator=(const MatrixFunctionAtomic&);
-
- void computeMu();
- bool taylorConverged(Index s, const MatrixType& F, const MatrixType& Fincr, const MatrixType& P);
-
- /** \brief Pointer to scalar function */
- StemFunction* m_f;
-
- /** \brief Size of matrix function */
- Index m_Arows;
-
- /** \brief Mean of eigenvalues */
- Scalar m_avgEival;
-
- /** \brief Argument shifted by mean of eigenvalues */
- MatrixType m_Ashifted;
-
- /** \brief Constant used to determine whether Taylor series has converged */
- RealScalar m_mu;
-};
-
-template <typename MatrixType>
-MatrixType MatrixFunctionAtomic<MatrixType>::compute(const MatrixType& A)
-{
- // TODO: Use that A is upper triangular
- m_Arows = A.rows();
- m_avgEival = A.trace() / Scalar(RealScalar(m_Arows));
- m_Ashifted = A - m_avgEival * MatrixType::Identity(m_Arows, m_Arows);
- computeMu();
- MatrixType F = m_f(m_avgEival, 0) * MatrixType::Identity(m_Arows, m_Arows);
- MatrixType P = m_Ashifted;
- MatrixType Fincr;
- for (Index s = 1; s < 1.1 * m_Arows + 10; s++) { // upper limit is fairly arbitrary
- Fincr = m_f(m_avgEival, static_cast<int>(s)) * P;
- F += Fincr;
- P = Scalar(RealScalar(1.0/(s + 1))) * P * m_Ashifted;
- if (taylorConverged(s, F, Fincr, P)) {
- return F;
- }
- }
- eigen_assert("Taylor series does not converge" && 0);
- return F;
-}
-
-/** \brief Compute \c m_mu. */
-template <typename MatrixType>
-void MatrixFunctionAtomic<MatrixType>::computeMu()
-{
- const MatrixType N = MatrixType::Identity(m_Arows, m_Arows) - m_Ashifted;
- VectorType e = VectorType::Ones(m_Arows);
- N.template triangularView<Upper>().solveInPlace(e);
- m_mu = e.cwiseAbs().maxCoeff();
-}
-
-/** \brief Determine whether Taylor series has converged */
-template <typename MatrixType>
-bool MatrixFunctionAtomic<MatrixType>::taylorConverged(Index s, const MatrixType& F,
- const MatrixType& Fincr, const MatrixType& P)
-{
- const Index n = F.rows();
- const RealScalar F_norm = F.cwiseAbs().rowwise().sum().maxCoeff();
- const RealScalar Fincr_norm = Fincr.cwiseAbs().rowwise().sum().maxCoeff();
- if (Fincr_norm < NumTraits<Scalar>::epsilon() * F_norm) {
- RealScalar delta = 0;
- RealScalar rfactorial = 1;
- for (Index r = 0; r < n; r++) {
- RealScalar mx = 0;
- for (Index i = 0; i < n; i++)
- mx = (std::max)(mx, std::abs(m_f(m_Ashifted(i, i) + m_avgEival, static_cast<int>(s+r))));
- if (r != 0)
- rfactorial *= RealScalar(r);
- delta = (std::max)(delta, mx / rfactorial);
- }
- const RealScalar P_norm = P.cwiseAbs().rowwise().sum().maxCoeff();
- if (m_mu * delta * P_norm < NumTraits<Scalar>::epsilon() * F_norm)
- return true;
- }
- return false;
-}
-
-} // end namespace Eigen
-
-#endif // EIGEN_MATRIX_FUNCTION_ATOMIC
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h b/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h
index c744fc0..cf5fffa 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h
@@ -1,7 +1,7 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
-// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
//
// This Source Code Form is subject to the terms of the Mozilla
@@ -11,91 +11,33 @@
#ifndef EIGEN_MATRIX_LOGARITHM
#define EIGEN_MATRIX_LOGARITHM
-#ifndef M_PI
-#define M_PI 3.141592653589793238462643383279503L
-#endif
-
namespace Eigen {
-/** \ingroup MatrixFunctions_Module
- * \class MatrixLogarithmAtomic
- * \brief Helper class for computing matrix logarithm of atomic matrices.
- *
- * \internal
- * Here, an atomic matrix is a triangular matrix whose diagonal
- * entries are close to each other.
- *
- * \sa class MatrixFunctionAtomic, MatrixBase::log()
- */
-template <typename MatrixType>
-class MatrixLogarithmAtomic
+namespace internal {
+
+template <typename Scalar>
+struct matrix_log_min_pade_degree
{
-public:
-
- typedef typename MatrixType::Scalar Scalar;
- // typedef typename MatrixType::Index Index;
- typedef typename NumTraits<Scalar>::Real RealScalar;
- // typedef typename internal::stem_function<Scalar>::type StemFunction;
- // typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
-
- /** \brief Constructor. */
- MatrixLogarithmAtomic() { }
-
- /** \brief Compute matrix logarithm of atomic matrix
- * \param[in] A argument of matrix logarithm, should be upper triangular and atomic
- * \returns The logarithm of \p A.
- */
- MatrixType compute(const MatrixType& A);
-
-private:
-
- void compute2x2(const MatrixType& A, MatrixType& result);
- void computeBig(const MatrixType& A, MatrixType& result);
- int getPadeDegree(float normTminusI);
- int getPadeDegree(double normTminusI);
- int getPadeDegree(long double normTminusI);
- void computePade(MatrixType& result, const MatrixType& T, int degree);
- void computePade3(MatrixType& result, const MatrixType& T);
- void computePade4(MatrixType& result, const MatrixType& T);
- void computePade5(MatrixType& result, const MatrixType& T);
- void computePade6(MatrixType& result, const MatrixType& T);
- void computePade7(MatrixType& result, const MatrixType& T);
- void computePade8(MatrixType& result, const MatrixType& T);
- void computePade9(MatrixType& result, const MatrixType& T);
- void computePade10(MatrixType& result, const MatrixType& T);
- void computePade11(MatrixType& result, const MatrixType& T);
-
- static const int minPadeDegree = 3;
- static const int maxPadeDegree = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
- std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
- std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
- std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
- 11; // quadruple precision
-
- // Prevent copying
- MatrixLogarithmAtomic(const MatrixLogarithmAtomic&);
- MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&);
+ static const int value = 3;
};
-/** \brief Compute logarithm of triangular matrix with clustered eigenvalues. */
-template <typename MatrixType>
-MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
+template <typename Scalar>
+struct matrix_log_max_pade_degree
{
- using std::log;
- MatrixType result(A.rows(), A.rows());
- if (A.rows() == 1)
- result(0,0) = log(A(0,0));
- else if (A.rows() == 2)
- compute2x2(A, result);
- else
- computeBig(A, result);
- return result;
-}
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
+ std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
+ std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
+ std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
+ 11; // quadruple precision
+};
/** \brief Compute logarithm of 2x2 triangular matrix. */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result)
+void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result)
{
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename MatrixType::RealScalar RealScalar;
using std::abs;
using std::ceil;
using std::imag;
@@ -108,59 +50,31 @@
result(1,0) = Scalar(0);
result(1,1) = logA11;
- if (A(0,0) == A(1,1)) {
+ Scalar y = A(1,1) - A(0,0);
+ if (y==Scalar(0))
+ {
result(0,1) = A(0,1) / A(0,0);
- } else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) {
- result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0));
- } else {
+ }
+ else if ((abs(A(0,0)) < RealScalar(0.5)*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1))))
+ {
+ result(0,1) = A(0,1) * (logA11 - logA00) / y;
+ }
+ else
+ {
// computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
- int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
- Scalar y = A(1,1) - A(0,0), x = A(1,1) + A(0,0);
- result(0,1) = A(0,1) * (Scalar(2) * numext::atanh2(y,x) + Scalar(0,2*M_PI*unwindingNumber)) / y;
+ int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI)));
+ result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,2*EIGEN_PI*unwindingNumber)) / y;
}
}
-/** \brief Compute logarithm of triangular matrices with size > 2.
- * \details This uses a inverse scale-and-square algorithm. */
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computeBig(const MatrixType& A, MatrixType& result)
-{
- using std::pow;
- int numberOfSquareRoots = 0;
- int numberOfExtraSquareRoots = 0;
- int degree;
- MatrixType T = A, sqrtT;
- const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1: // single precision
- maxPadeDegree<= 7? 2.6429608311114350e-1: // double precision
- maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
- maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
- 1.1880960220216759245467951592883642e-1L; // quadruple precision
-
- while (true) {
- RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
- if (normTminusI < maxNormForPade) {
- degree = getPadeDegree(normTminusI);
- int degree2 = getPadeDegree(normTminusI / RealScalar(2));
- if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
- break;
- ++numberOfExtraSquareRoots;
- }
- MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
- T = sqrtT.template triangularView<Upper>();
- ++numberOfSquareRoots;
- }
-
- computePade(result, T, degree);
- result *= pow(RealScalar(2), numberOfSquareRoots);
-}
-
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
-template <typename MatrixType>
-int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI)
+inline int matrix_log_get_pade_degree(float normTminusI)
{
const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
5.3149729967117310e-1 };
- int degree = 3;
+ const int minPadeDegree = matrix_log_min_pade_degree<float>::value;
+ const int maxPadeDegree = matrix_log_max_pade_degree<float>::value;
+ int degree = minPadeDegree;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
@@ -168,12 +82,13 @@
}
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
-template <typename MatrixType>
-int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI)
+inline int matrix_log_get_pade_degree(double normTminusI)
{
const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
- int degree = 3;
+ const int minPadeDegree = matrix_log_min_pade_degree<double>::value;
+ const int maxPadeDegree = matrix_log_max_pade_degree<double>::value;
+ int degree = minPadeDegree;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
@@ -181,8 +96,7 @@
}
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
-template <typename MatrixType>
-int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
+inline int matrix_log_get_pade_degree(long double normTminusI)
{
#if LDBL_MANT_DIG == 53 // double precision
const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
@@ -204,7 +118,9 @@
3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
#endif
- int degree = 3;
+ const int minPadeDegree = matrix_log_min_pade_degree<long double>::value;
+ const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value;
+ int degree = minPadeDegree;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
@@ -213,197 +129,168 @@
/* \brief Compute Pade approximation to matrix logarithm */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result, const MatrixType& T, int degree)
+void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree)
{
- switch (degree) {
- case 3: computePade3(result, T); break;
- case 4: computePade4(result, T); break;
- case 5: computePade5(result, T); break;
- case 6: computePade6(result, T); break;
- case 7: computePade7(result, T); break;
- case 8: computePade8(result, T); break;
- case 9: computePade9(result, T); break;
- case 10: computePade10(result, T); break;
- case 11: computePade11(result, T); break;
- default: assert(false); // should never happen
+ typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
+ const int minPadeDegree = 3;
+ const int maxPadeDegree = 11;
+ assert(degree >= minPadeDegree && degree <= maxPadeDegree);
+
+ const RealScalar nodes[][maxPadeDegree] = {
+ { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3
+ 0.8872983346207416885179265399782400L },
+ { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4
+ 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L },
+ { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5
+ 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
+ 0.9530899229693319963988134391496965L },
+ { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6
+ 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
+ 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L },
+ { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7
+ 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
+ 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
+ 0.9745539561713792622630948420239256L },
+ { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8
+ 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
+ 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
+ 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L },
+ { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9
+ 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
+ 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
+ 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
+ 0.9840801197538130449177881014518364L },
+ { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10
+ 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
+ 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
+ 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
+ 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L },
+ { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11
+ 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
+ 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
+ 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
+ 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
+ 0.9891143290730284964019690005614287L } };
+
+ const RealScalar weights[][maxPadeDegree] = {
+ { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3
+ 0.2777777777777777777777777777777778L },
+ { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4
+ 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L },
+ { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5
+ 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
+ 0.1184634425280945437571320203599587L },
+ { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6
+ 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
+ 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L },
+ { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7
+ 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
+ 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
+ 0.0647424830844348466353057163395410L },
+ { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8
+ 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
+ 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
+ 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L },
+ { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9
+ 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
+ 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
+ 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
+ 0.0406371941807872059859460790552618L },
+ { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10
+ 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
+ 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
+ 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
+ 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L },
+ { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11
+ 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
+ 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
+ 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
+ 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
+ 0.0278342835580868332413768602212743L } };
+
+ MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
+ result.setZero(T.rows(), T.rows());
+ for (int k = 0; k < degree; ++k) {
+ RealScalar weight = weights[degree-minPadeDegree][k];
+ RealScalar node = nodes[degree-minPadeDegree][k];
+ result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI)
+ .template triangularView<Upper>().solve(TminusI);
}
}
+/** \brief Compute logarithm of triangular matrices with size > 2.
+ * \details This uses a inverse scale-and-square algorithm. */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result, const MatrixType& T)
+void matrix_log_compute_big(const MatrixType& A, MatrixType& result)
{
- const int degree = 3;
- const RealScalar nodes[] = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L,
- 0.8872983346207416885179265399782400L };
- const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L,
- 0.2777777777777777777777777777777778L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ using std::pow;
+
+ int numberOfSquareRoots = 0;
+ int numberOfExtraSquareRoots = 0;
+ int degree;
+ MatrixType T = A, sqrtT;
+
+ int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value;
+ const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1L: // single precision
+ maxPadeDegree<= 7? 2.6429608311114350e-1L: // double precision
+ maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
+ maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
+ 1.1880960220216759245467951592883642e-1L; // quadruple precision
+
+ while (true) {
+ RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
+ if (normTminusI < maxNormForPade) {
+ degree = matrix_log_get_pade_degree(normTminusI);
+ int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2));
+ if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
+ break;
+ ++numberOfExtraSquareRoots;
+ }
+ matrix_sqrt_triangular(T, sqrtT);
+ T = sqrtT.template triangularView<Upper>();
+ ++numberOfSquareRoots;
+ }
+
+ matrix_log_compute_pade(result, T, degree);
+ result *= pow(RealScalar(2), numberOfSquareRoots);
}
+/** \ingroup MatrixFunctions_Module
+ * \class MatrixLogarithmAtomic
+ * \brief Helper class for computing matrix logarithm of atomic matrices.
+ *
+ * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other.
+ *
+ * \sa class MatrixFunctionAtomic, MatrixBase::log()
+ */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T)
+class MatrixLogarithmAtomic
{
- const int degree = 4;
- const RealScalar nodes[] = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L,
- 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L };
- const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L,
- 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+public:
+ /** \brief Compute matrix logarithm of atomic matrix
+ * \param[in] A argument of matrix logarithm, should be upper triangular and atomic
+ * \returns The logarithm of \p A.
+ */
+ MatrixType compute(const MatrixType& A);
+};
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result, const MatrixType& T)
+MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
{
- const int degree = 5;
- const RealScalar nodes[] = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L,
- 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
- 0.9530899229693319963988134391496965L };
- const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L,
- 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
- 0.1184634425280945437571320203599587L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
+ using std::log;
+ MatrixType result(A.rows(), A.rows());
+ if (A.rows() == 1)
+ result(0,0) = log(A(0,0));
+ else if (A.rows() == 2)
+ matrix_log_compute_2x2(A, result);
+ else
+ matrix_log_compute_big(A, result);
+ return result;
}
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result, const MatrixType& T)
-{
- const int degree = 6;
- const RealScalar nodes[] = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L,
- 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
- 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L };
- const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L,
- 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
- 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
-
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result, const MatrixType& T)
-{
- const int degree = 7;
- const RealScalar nodes[] = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L,
- 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
- 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
- 0.9745539561713792622630948420239256L };
- const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L,
- 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
- 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
- 0.0647424830844348466353057163395410L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
-
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result, const MatrixType& T)
-{
- const int degree = 8;
- const RealScalar nodes[] = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L,
- 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
- 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
- 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L };
- const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L,
- 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
- 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
- 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
-
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T)
-{
- const int degree = 9;
- const RealScalar nodes[] = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L,
- 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
- 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
- 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
- 0.9840801197538130449177881014518364L };
- const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L,
- 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
- 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
- 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
- 0.0406371941807872059859460790552618L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
-
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T)
-{
- const int degree = 10;
- const RealScalar nodes[] = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L,
- 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
- 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
- 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
- 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L };
- const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L,
- 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
- 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
- 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
- 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
-
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result, const MatrixType& T)
-{
- const int degree = 11;
- const RealScalar nodes[] = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L,
- 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
- 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
- 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
- 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
- 0.9891143290730284964019690005614287L };
- const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L,
- 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
- 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
- 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
- 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
- 0.0278342835580868332413768602212743L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+} // end of namespace internal
/** \ingroup MatrixFunctions_Module
*
@@ -421,45 +308,45 @@
: public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
{
public:
-
typedef typename Derived::Scalar Scalar;
typedef typename Derived::Index Index;
+protected:
+ typedef typename internal::ref_selector<Derived>::type DerivedNested;
+
+public:
+
/** \brief Constructor.
*
* \param[in] A %Matrix (expression) forming the argument of the matrix logarithm.
*/
- MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
+ explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
/** \brief Compute the matrix logarithm.
*
- * \param[out] result Logarithm of \p A, where \A is as specified in the constructor.
+ * \param[out] result Logarithm of \c A, where \c A is as specified in the constructor.
*/
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
- typedef typename Derived::PlainObject PlainObject;
- typedef internal::traits<PlainObject> Traits;
+ typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
+ typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
+ typedef internal::traits<DerivedEvalTypeClean> Traits;
static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
- static const int Options = PlainObject::Options;
typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
- typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
- typedef MatrixLogarithmAtomic<DynMatrixType> AtomicType;
+ typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
+ typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType;
AtomicType atomic;
- const PlainObject Aevaluated = m_A.eval();
- MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
- mf.compute(result);
+ internal::matrix_function_compute<typename DerivedEvalTypeClean::PlainObject>::run(m_A, atomic, result);
}
Index rows() const { return m_A.rows(); }
Index cols() const { return m_A.cols(); }
private:
- typename internal::nested<Derived>::type m_A;
-
- MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&);
+ const DerivedNested m_A;
};
namespace internal {
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
index 78a307e..a3273da 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
@@ -14,19 +14,51 @@
template<typename MatrixType> class MatrixPower;
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Proxy for the matrix power of some matrix.
+ *
+ * \tparam MatrixType type of the base, a matrix.
+ *
+ * This class holds the arguments to the matrix power until it is
+ * assigned or evaluated for some other reason (so the argument
+ * should not be changed in the meantime). It is the return type of
+ * MatrixPower::operator() and related functions and most of the
+ * time this is the only way it is used.
+ */
+/* TODO This class is only used by MatrixPower, so it should be nested
+ * into MatrixPower, like MatrixPower::ReturnValue. However, my
+ * compiler complained about unused template parameter in the
+ * following declaration in namespace internal.
+ *
+ * template<typename MatrixType>
+ * struct traits<MatrixPower<MatrixType>::ReturnValue>;
+ */
template<typename MatrixType>
-class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
+class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> >
{
public:
typedef typename MatrixType::RealScalar RealScalar;
typedef typename MatrixType::Index Index;
- MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
+ /**
+ * \brief Constructor.
+ *
+ * \param[in] pow %MatrixPower storing the base.
+ * \param[in] p scalar, the exponent of the matrix power.
+ */
+ MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
{ }
+ /**
+ * \brief Compute the matrix power.
+ *
+ * \param[out] result
+ */
template<typename ResultType>
- inline void evalTo(ResultType& res) const
- { m_pow.compute(res, m_p); }
+ inline void evalTo(ResultType& result) const
+ { m_pow.compute(result, m_p); }
Index rows() const { return m_pow.rows(); }
Index cols() const { return m_pow.cols(); }
@@ -34,11 +66,25 @@
private:
MatrixPower<MatrixType>& m_pow;
const RealScalar m_p;
- MatrixPowerRetval& operator=(const MatrixPowerRetval&);
};
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Class for computing matrix powers.
+ *
+ * \tparam MatrixType type of the base, expected to be an instantiation
+ * of the Matrix class template.
+ *
+ * This class is capable of computing triangular real/complex matrices
+ * raised to a power in the interval \f$ (-1, 1) \f$.
+ *
+ * \note Currently this class is only used by MatrixPower. One may
+ * insist that this be nested into MatrixPower. This class is here to
+ * faciliate future development of triangular matrix functions.
+ */
template<typename MatrixType>
-class MatrixPowerAtomic
+class MatrixPowerAtomic : internal::noncopyable
{
private:
enum {
@@ -49,14 +95,14 @@
typedef typename MatrixType::RealScalar RealScalar;
typedef std::complex<RealScalar> ComplexScalar;
typedef typename MatrixType::Index Index;
- typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;
+ typedef Block<MatrixType,Dynamic,Dynamic> ResultType;
const MatrixType& m_A;
RealScalar m_p;
- void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const;
- void compute2x2(MatrixType& res, RealScalar p) const;
- void computeBig(MatrixType& res) const;
+ void computePade(int degree, const MatrixType& IminusT, ResultType& res) const;
+ void compute2x2(ResultType& res, RealScalar p) const;
+ void computeBig(ResultType& res) const;
static int getPadeDegree(float normIminusT);
static int getPadeDegree(double normIminusT);
static int getPadeDegree(long double normIminusT);
@@ -64,24 +110,45 @@
static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
public:
+ /**
+ * \brief Constructor.
+ *
+ * \param[in] T the base of the matrix power.
+ * \param[in] p the exponent of the matrix power, should be in
+ * \f$ (-1, 1) \f$.
+ *
+ * The class stores a reference to T, so it should not be changed
+ * (or destroyed) before evaluation. Only the upper triangular
+ * part of T is read.
+ */
MatrixPowerAtomic(const MatrixType& T, RealScalar p);
- void compute(MatrixType& res) const;
+
+ /**
+ * \brief Compute the matrix power.
+ *
+ * \param[out] res \f$ A^p \f$ where A and p are specified in the
+ * constructor.
+ */
+ void compute(ResultType& res) const;
};
template<typename MatrixType>
MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
m_A(T), m_p(p)
-{ eigen_assert(T.rows() == T.cols()); }
+{
+ eigen_assert(T.rows() == T.cols());
+ eigen_assert(p > -1 && p < 1);
+}
template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
+void MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const
{
- res.resizeLike(m_A);
+ using std::pow;
switch (m_A.rows()) {
case 0:
break;
case 1:
- res(0,0) = std::pow(m_A(0,0), m_p);
+ res(0,0) = pow(m_A(0,0), m_p);
break;
case 2:
compute2x2(res, m_p);
@@ -92,24 +159,24 @@
}
template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const
+void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const
{
- int i = degree<<1;
- res = (m_p-degree) / ((i-1)<<1) * IminusT;
+ int i = 2*degree;
+ res = (m_p-degree) / (2*i-2) * IminusT;
+
for (--i; i; --i) {
res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
- .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval();
+ .solve((i==1 ? -m_p : i&1 ? (-m_p-i/2)/(2*i) : (m_p-i/2)/(2*i-2)) * IminusT).eval();
}
res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
}
// This function assumes that res has the correct size (see bug 614)
template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const
+void MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const
{
using std::abs;
using std::pow;
-
res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);
for (Index i=1; i < m_A.cols(); ++i) {
@@ -125,32 +192,20 @@
}
template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
+void MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const
{
+ using std::ldexp;
const int digits = std::numeric_limits<RealScalar>::digits;
- const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
- digits <= 53? 2.789358995219730e-1: // double precision
- digits <= 64? 2.4471944416607995472e-1L: // extended precision
- digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double
- 9.134603732914548552537150753385375e-2L; // quadruple precision
+ const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1L // single precision
+ : digits <= 53? 2.789358995219730e-1L // double precision
+ : digits <= 64? 2.4471944416607995472e-1L // extended precision
+ : digits <= 106? 1.1016843812851143391275867258512e-1L // double-double
+ : 9.134603732914548552537150753385375e-2L; // quadruple precision
MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
RealScalar normIminusT;
int degree, degree2, numberOfSquareRoots = 0;
bool hasExtraSquareRoot = false;
- /* FIXME
- * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite
- * loop. We should move 0 eigenvalues to bottom right corner. We need not
- * worry about tiny values (e.g. 1e-300) because they will reach 1 if
- * repetitively sqrt'ed.
- *
- * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the
- * bottom right corner.
- *
- * [ T A ]^p [ T^p (T^-1 T^p A) ]
- * [ ] = [ ]
- * [ 0 0 ] [ 0 0 ]
- */
for (Index i=0; i < m_A.cols(); ++i)
eigen_assert(m_A(i,i) != RealScalar(0));
@@ -164,14 +219,14 @@
break;
hasExtraSquareRoot = true;
}
- MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
+ matrix_sqrt_triangular(T, sqrtT);
T = sqrtT.template triangularView<Upper>();
++numberOfSquareRoots;
}
computePade(degree, IminusT, res);
for (; numberOfSquareRoots; --numberOfSquareRoots) {
- compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));
+ compute2x2(res, ldexp(m_p, -numberOfSquareRoots));
res = res.template triangularView<Upper>() * res;
}
compute2x2(res, m_p);
@@ -209,7 +264,7 @@
1.999045567181744e-1L, 2.789358995219730e-1L };
#elif LDBL_MANT_DIG <= 64
const int maxPadeDegree = 8;
- const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
+ const long double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
#elif LDBL_MANT_DIG <= 106
const int maxPadeDegree = 10;
@@ -236,19 +291,28 @@
inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
{
- ComplexScalar logCurr = std::log(curr);
- ComplexScalar logPrev = std::log(prev);
- int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
- ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);
- return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev);
+ using std::ceil;
+ using std::exp;
+ using std::log;
+ using std::sinh;
+
+ ComplexScalar logCurr = log(curr);
+ ComplexScalar logPrev = log(prev);
+ int unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI));
+ ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, EIGEN_PI*unwindingNumber);
+ return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev);
}
template<typename MatrixType>
inline typename MatrixPowerAtomic<MatrixType>::RealScalar
MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
{
- RealScalar w = numext::atanh2(curr - prev, curr + prev);
- return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev);
+ using std::exp;
+ using std::log;
+ using std::sinh;
+
+ RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2);
+ return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev);
}
/**
@@ -271,15 +335,9 @@
* Output: \verbinclude MatrixPower_optimal.out
*/
template<typename MatrixType>
-class MatrixPower
+class MatrixPower : internal::noncopyable
{
private:
- enum {
- RowsAtCompileTime = MatrixType::RowsAtCompileTime,
- ColsAtCompileTime = MatrixType::ColsAtCompileTime,
- MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
- MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
- };
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
typedef typename MatrixType::Index Index;
@@ -293,7 +351,11 @@
* The class stores a reference to A, so it should not be changed
* (or destroyed) before evaluation.
*/
- explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0)
+ explicit MatrixPower(const MatrixType& A) :
+ m_A(A),
+ m_conditionNumber(0),
+ m_rank(A.cols()),
+ m_nulls(0)
{ eigen_assert(A.rows() == A.cols()); }
/**
@@ -303,8 +365,8 @@
* \return The expression \f$ A^p \f$, where A is specified in the
* constructor.
*/
- const MatrixPowerRetval<MatrixType> operator()(RealScalar p)
- { return MatrixPowerRetval<MatrixType>(*this, p); }
+ const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p)
+ { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); }
/**
* \brief Compute the matrix power.
@@ -321,21 +383,54 @@
private:
typedef std::complex<RealScalar> ComplexScalar;
- typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,
- MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;
+ typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0,
+ MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix;
+ /** \brief Reference to the base of matrix power. */
typename MatrixType::Nested m_A;
+
+ /** \brief Temporary storage. */
MatrixType m_tmp;
- ComplexMatrix m_T, m_U, m_fT;
+
+ /** \brief Store the result of Schur decomposition. */
+ ComplexMatrix m_T, m_U;
+
+ /** \brief Store fractional power of m_T. */
+ ComplexMatrix m_fT;
+
+ /**
+ * \brief Condition number of m_A.
+ *
+ * It is initialized as 0 to avoid performing unnecessary Schur
+ * decomposition, which is the bottleneck.
+ */
RealScalar m_conditionNumber;
- RealScalar modfAndInit(RealScalar, RealScalar*);
+ /** \brief Rank of m_A. */
+ Index m_rank;
+
+ /** \brief Rank deficiency of m_A. */
+ Index m_nulls;
+
+ /**
+ * \brief Split p into integral part and fractional part.
+ *
+ * \param[in] p The exponent.
+ * \param[out] p The fractional part ranging in \f$ (-1, 1) \f$.
+ * \param[out] intpart The integral part.
+ *
+ * Only if the fractional part is nonzero, it calls initialize().
+ */
+ void split(RealScalar& p, RealScalar& intpart);
+
+ /** \brief Perform Schur decomposition for fractional power. */
+ void initialize();
template<typename ResultType>
- void computeIntPower(ResultType&, RealScalar);
+ void computeIntPower(ResultType& res, RealScalar p);
template<typename ResultType>
- void computeFracPower(ResultType&, RealScalar);
+ void computeFracPower(ResultType& res, RealScalar p);
template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
static void revertSchur(
@@ -354,59 +449,102 @@
template<typename ResultType>
void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
{
+ using std::pow;
switch (cols()) {
case 0:
break;
case 1:
- res(0,0) = std::pow(m_A.coeff(0,0), p);
+ res(0,0) = pow(m_A.coeff(0,0), p);
break;
default:
- RealScalar intpart, x = modfAndInit(p, &intpart);
+ RealScalar intpart;
+ split(p, intpart);
+
+ res = MatrixType::Identity(rows(), cols());
computeIntPower(res, intpart);
- computeFracPower(res, x);
+ if (p) computeFracPower(res, p);
}
}
template<typename MatrixType>
-typename MatrixPower<MatrixType>::RealScalar
-MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
+void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart)
{
- typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;
+ using std::floor;
+ using std::pow;
- *intpart = std::floor(x);
- RealScalar res = x - *intpart;
+ intpart = floor(p);
+ p -= intpart;
- if (!m_conditionNumber && res) {
- const ComplexSchur<MatrixType> schurOfA(m_A);
- m_T = schurOfA.matrixT();
- m_U = schurOfA.matrixU();
-
- const RealArray absTdiag = m_T.diagonal().array().abs();
- m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
+ // Perform Schur decomposition if it is not yet performed and the power is
+ // not an integer.
+ if (!m_conditionNumber && p)
+ initialize();
+
+ // Choose the more stable of intpart = floor(p) and intpart = ceil(p).
+ if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) {
+ --p;
+ ++intpart;
+ }
+}
+
+template<typename MatrixType>
+void MatrixPower<MatrixType>::initialize()
+{
+ const ComplexSchur<MatrixType> schurOfA(m_A);
+ JacobiRotation<ComplexScalar> rot;
+ ComplexScalar eigenvalue;
+
+ m_fT.resizeLike(m_A);
+ m_T = schurOfA.matrixT();
+ m_U = schurOfA.matrixU();
+ m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff();
+
+ // Move zero eigenvalues to the bottom right corner.
+ for (Index i = cols()-1; i>=0; --i) {
+ if (m_rank <= 2)
+ return;
+ if (m_T.coeff(i,i) == RealScalar(0)) {
+ for (Index j=i+1; j < m_rank; ++j) {
+ eigenvalue = m_T.coeff(j,j);
+ rot.makeGivens(m_T.coeff(j-1,j), eigenvalue);
+ m_T.applyOnTheRight(j-1, j, rot);
+ m_T.applyOnTheLeft(j-1, j, rot.adjoint());
+ m_T.coeffRef(j-1,j-1) = eigenvalue;
+ m_T.coeffRef(j,j) = RealScalar(0);
+ m_U.applyOnTheRight(j-1, j, rot);
+ }
+ --m_rank;
+ }
}
- if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
- --res;
- ++*intpart;
+ m_nulls = rows() - m_rank;
+ if (m_nulls) {
+ eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero()
+ && "Base of matrix power should be invertible or with a semisimple zero eigenvalue.");
+ m_fT.bottomRows(m_nulls).fill(RealScalar(0));
}
- return res;
}
template<typename MatrixType>
template<typename ResultType>
void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
{
- RealScalar pp = std::abs(p);
+ using std::abs;
+ using std::fmod;
+ RealScalar pp = abs(p);
- if (p<0) m_tmp = m_A.inverse();
- else m_tmp = m_A;
+ if (p<0)
+ m_tmp = m_A.inverse();
+ else
+ m_tmp = m_A;
- res = MatrixType::Identity(rows(), cols());
- while (pp >= 1) {
- if (std::fmod(pp, 2) >= 1)
+ while (true) {
+ if (fmod(pp, 2) >= 1)
res = m_tmp * res;
- m_tmp *= m_tmp;
pp /= 2;
+ if (pp < 1)
+ break;
+ m_tmp *= m_tmp;
}
}
@@ -414,12 +552,17 @@
template<typename ResultType>
void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
{
- if (p) {
- eigen_assert(m_conditionNumber);
- MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);
- revertSchur(m_tmp, m_fT, m_U);
- res = m_tmp * res;
+ Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank);
+ eigen_assert(m_conditionNumber);
+ eigen_assert(m_rank + m_nulls == rows());
+
+ MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp);
+ if (m_nulls) {
+ m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>()
+ .solve(blockTp * m_T.topRightCorner(m_rank, m_nulls));
}
+ revertSchur(m_tmp, m_fT, m_U);
+ res = m_tmp * res;
}
template<typename MatrixType>
@@ -463,7 +606,7 @@
* \brief Constructor.
*
* \param[in] A %Matrix (expression), the base of the matrix power.
- * \param[in] p scalar, the exponent of the matrix power.
+ * \param[in] p real scalar, the exponent of the matrix power.
*/
MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
{ }
@@ -475,8 +618,8 @@
* constructor.
*/
template<typename ResultType>
- inline void evalTo(ResultType& res) const
- { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }
+ inline void evalTo(ResultType& result) const
+ { MatrixPower<PlainObject>(m_A.eval()).compute(result, m_p); }
Index rows() const { return m_A.rows(); }
Index cols() const { return m_A.cols(); }
@@ -484,25 +627,83 @@
private:
const Derived& m_A;
const RealScalar m_p;
- MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
+};
+
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Proxy for the matrix power of some matrix (expression).
+ *
+ * \tparam Derived type of the base, a matrix (expression).
+ *
+ * This class holds the arguments to the matrix power until it is
+ * assigned or evaluated for some other reason (so the argument
+ * should not be changed in the meantime). It is the return type of
+ * MatrixBase::pow() and related functions and most of the
+ * time this is the only way it is used.
+ */
+template<typename Derived>
+class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> >
+{
+ public:
+ typedef typename Derived::PlainObject PlainObject;
+ typedef typename std::complex<typename Derived::RealScalar> ComplexScalar;
+ typedef typename Derived::Index Index;
+
+ /**
+ * \brief Constructor.
+ *
+ * \param[in] A %Matrix (expression), the base of the matrix power.
+ * \param[in] p complex scalar, the exponent of the matrix power.
+ */
+ MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p)
+ { }
+
+ /**
+ * \brief Compute the matrix power.
+ *
+ * Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$
+ * \exp(p \log(A)) \f$.
+ *
+ * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the
+ * constructor.
+ */
+ template<typename ResultType>
+ inline void evalTo(ResultType& result) const
+ { result = (m_p * m_A.log()).exp(); }
+
+ Index rows() const { return m_A.rows(); }
+ Index cols() const { return m_A.cols(); }
+
+ private:
+ const Derived& m_A;
+ const ComplexScalar m_p;
};
namespace internal {
template<typename MatrixPowerType>
-struct traits< MatrixPowerRetval<MatrixPowerType> >
+struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> >
{ typedef typename MatrixPowerType::PlainObject ReturnType; };
template<typename Derived>
struct traits< MatrixPowerReturnValue<Derived> >
{ typedef typename Derived::PlainObject ReturnType; };
+template<typename Derived>
+struct traits< MatrixComplexPowerReturnValue<Derived> >
+{ typedef typename Derived::PlainObject ReturnType; };
+
}
template<typename Derived>
const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
{ return MatrixPowerReturnValue<Derived>(derived(), p); }
+template<typename Derived>
+const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const
+{ return MatrixComplexPowerReturnValue<Derived>(derived(), p); }
+
} // namespace Eigen
#endif // EIGEN_MATRIX_POWER
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixSquareRoot.h b/unsupported/Eigen/src/MatrixFunctions/MatrixSquareRoot.h
index b48ea9d..2e5abda 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixSquareRoot.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixSquareRoot.h
@@ -1,7 +1,7 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
-// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
@@ -12,133 +12,16 @@
namespace Eigen {
-/** \ingroup MatrixFunctions_Module
- * \brief Class for computing matrix square roots of upper quasi-triangular matrices.
- * \tparam MatrixType type of the argument of the matrix square root,
- * expected to be an instantiation of the Matrix class template.
- *
- * This class computes the square root of the upper quasi-triangular
- * matrix stored in the upper Hessenberg part of the matrix passed to
- * the constructor.
- *
- * \sa MatrixSquareRoot, MatrixSquareRootTriangular
- */
-template <typename MatrixType>
-class MatrixSquareRootQuasiTriangular
-{
- public:
-
- /** \brief Constructor.
- *
- * \param[in] A upper quasi-triangular matrix whose square root
- * is to be computed.
- *
- * The class stores a reference to \p A, so it should not be
- * changed (or destroyed) before compute() is called.
- */
- MatrixSquareRootQuasiTriangular(const MatrixType& A)
- : m_A(A)
- {
- eigen_assert(A.rows() == A.cols());
- }
-
- /** \brief Compute the matrix square root
- *
- * \param[out] result square root of \p A, as specified in the constructor.
- *
- * Only the upper Hessenberg part of \p result is updated, the
- * rest is not touched. See MatrixBase::sqrt() for details on
- * how this computation is implemented.
- */
- template <typename ResultType> void compute(ResultType &result);
-
- private:
- typedef typename MatrixType::Index Index;
- typedef typename MatrixType::Scalar Scalar;
-
- void computeDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
- void computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
- void compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i);
- void compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j);
- void compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j);
- void compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j);
- void compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j);
-
- template <typename SmallMatrixType>
- static void solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A,
- const SmallMatrixType& B, const SmallMatrixType& C);
-
- const MatrixType& m_A;
-};
-
-template <typename MatrixType>
-template <typename ResultType>
-void MatrixSquareRootQuasiTriangular<MatrixType>::compute(ResultType &result)
-{
- result.resize(m_A.rows(), m_A.cols());
- computeDiagonalPartOfSqrt(result, m_A);
- computeOffDiagonalPartOfSqrt(result, m_A);
-}
-
-// pre: T is quasi-upper-triangular and sqrtT is a zero matrix of the same size
-// post: the diagonal blocks of sqrtT are the square roots of the diagonal blocks of T
-template <typename MatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>::computeDiagonalPartOfSqrt(MatrixType& sqrtT,
- const MatrixType& T)
-{
- using std::sqrt;
- const Index size = m_A.rows();
- for (Index i = 0; i < size; i++) {
- if (i == size - 1 || T.coeff(i+1, i) == 0) {
- eigen_assert(T(i,i) >= 0);
- sqrtT.coeffRef(i,i) = sqrt(T.coeff(i,i));
- }
- else {
- compute2x2diagonalBlock(sqrtT, T, i);
- ++i;
- }
- }
-}
-
-// pre: T is quasi-upper-triangular and diagonal blocks of sqrtT are square root of diagonal blocks of T.
-// post: sqrtT is the square root of T.
-template <typename MatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>::computeOffDiagonalPartOfSqrt(MatrixType& sqrtT,
- const MatrixType& T)
-{
- const Index size = m_A.rows();
- for (Index j = 1; j < size; j++) {
- if (T.coeff(j, j-1) != 0) // if T(j-1:j, j-1:j) is a 2-by-2 block
- continue;
- for (Index i = j-1; i >= 0; i--) {
- if (i > 0 && T.coeff(i, i-1) != 0) // if T(i-1:i, i-1:i) is a 2-by-2 block
- continue;
- bool iBlockIs2x2 = (i < size - 1) && (T.coeff(i+1, i) != 0);
- bool jBlockIs2x2 = (j < size - 1) && (T.coeff(j+1, j) != 0);
- if (iBlockIs2x2 && jBlockIs2x2)
- compute2x2offDiagonalBlock(sqrtT, T, i, j);
- else if (iBlockIs2x2 && !jBlockIs2x2)
- compute2x1offDiagonalBlock(sqrtT, T, i, j);
- else if (!iBlockIs2x2 && jBlockIs2x2)
- compute1x2offDiagonalBlock(sqrtT, T, i, j);
- else if (!iBlockIs2x2 && !jBlockIs2x2)
- compute1x1offDiagonalBlock(sqrtT, T, i, j);
- }
- }
-}
+namespace internal {
// pre: T.block(i,i,2,2) has complex conjugate eigenvalues
// post: sqrtT.block(i,i,2,2) is square root of T.block(i,i,2,2)
-template <typename MatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>
- ::compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i)
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular_2x2_diagonal_block(const MatrixType& T, typename MatrixType::Index i, ResultType& sqrtT)
{
// TODO: This case (2-by-2 blocks with complex conjugate eigenvalues) is probably hidden somewhere
// in EigenSolver. If we expose it, we could call it directly from here.
+ typedef typename traits<MatrixType>::Scalar Scalar;
Matrix<Scalar,2,2> block = T.template block<2,2>(i,i);
EigenSolver<Matrix<Scalar,2,2> > es(block);
sqrtT.template block<2,2>(i,i)
@@ -148,21 +31,19 @@
// pre: block structure of T is such that (i,j) is a 1x1 block,
// all blocks of sqrtT to left of and below (i,j) are correct
// post: sqrtT(i,j) has the correct value
-template <typename MatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>
- ::compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j)
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular_1x1_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
{
+ typedef typename traits<MatrixType>::Scalar Scalar;
Scalar tmp = (sqrtT.row(i).segment(i+1,j-i-1) * sqrtT.col(j).segment(i+1,j-i-1)).value();
sqrtT.coeffRef(i,j) = (T.coeff(i,j) - tmp) / (sqrtT.coeff(i,i) + sqrtT.coeff(j,j));
}
// similar to compute1x1offDiagonalBlock()
-template <typename MatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>
- ::compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j)
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular_1x2_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
{
+ typedef typename traits<MatrixType>::Scalar Scalar;
Matrix<Scalar,1,2> rhs = T.template block<1,2>(i,j);
if (j-i > 1)
rhs -= sqrtT.block(i, i+1, 1, j-i-1) * sqrtT.block(i+1, j, j-i-1, 2);
@@ -172,11 +53,10 @@
}
// similar to compute1x1offDiagonalBlock()
-template <typename MatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>
- ::compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j)
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular_2x1_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
{
+ typedef typename traits<MatrixType>::Scalar Scalar;
Matrix<Scalar,2,1> rhs = T.template block<2,1>(i,j);
if (j-i > 2)
rhs -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 1);
@@ -185,32 +65,11 @@
sqrtT.template block<2,1>(i,j) = A.fullPivLu().solve(rhs);
}
-// similar to compute1x1offDiagonalBlock()
-template <typename MatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>
- ::compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
- typename MatrixType::Index i, typename MatrixType::Index j)
-{
- Matrix<Scalar,2,2> A = sqrtT.template block<2,2>(i,i);
- Matrix<Scalar,2,2> B = sqrtT.template block<2,2>(j,j);
- Matrix<Scalar,2,2> C = T.template block<2,2>(i,j);
- if (j-i > 2)
- C -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 2);
- Matrix<Scalar,2,2> X;
- solveAuxiliaryEquation(X, A, B, C);
- sqrtT.template block<2,2>(i,j) = X;
-}
-
// solves the equation A X + X B = C where all matrices are 2-by-2
template <typename MatrixType>
-template <typename SmallMatrixType>
-void MatrixSquareRootQuasiTriangular<MatrixType>
- ::solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A,
- const SmallMatrixType& B, const SmallMatrixType& C)
+void matrix_sqrt_quasi_triangular_solve_auxiliary_equation(MatrixType& X, const MatrixType& A, const MatrixType& B, const MatrixType& C)
{
- EIGEN_STATIC_ASSERT((internal::is_same<SmallMatrixType, Matrix<Scalar,2,2> >::value),
- EIGEN_INTERNAL_ERROR_PLEASE_FILE_A_BUG_REPORT);
-
+ typedef typename traits<MatrixType>::Scalar Scalar;
Matrix<Scalar,4,4> coeffMatrix = Matrix<Scalar,4,4>::Zero();
coeffMatrix.coeffRef(0,0) = A.coeff(0,0) + B.coeff(0,0);
coeffMatrix.coeffRef(1,1) = A.coeff(0,0) + B.coeff(1,1);
@@ -224,13 +83,13 @@
coeffMatrix.coeffRef(2,3) = B.coeff(1,0);
coeffMatrix.coeffRef(3,1) = A.coeff(1,0);
coeffMatrix.coeffRef(3,2) = B.coeff(0,1);
-
+
Matrix<Scalar,4,1> rhs;
rhs.coeffRef(0) = C.coeff(0,0);
rhs.coeffRef(1) = C.coeff(0,1);
rhs.coeffRef(2) = C.coeff(1,0);
rhs.coeffRef(3) = C.coeff(1,1);
-
+
Matrix<Scalar,4,1> result;
result = coeffMatrix.fullPivLu().solve(rhs);
@@ -240,165 +99,205 @@
X.coeffRef(1,1) = result.coeff(3);
}
+// similar to compute1x1offDiagonalBlock()
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular_2x2_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
+{
+ typedef typename traits<MatrixType>::Scalar Scalar;
+ Matrix<Scalar,2,2> A = sqrtT.template block<2,2>(i,i);
+ Matrix<Scalar,2,2> B = sqrtT.template block<2,2>(j,j);
+ Matrix<Scalar,2,2> C = T.template block<2,2>(i,j);
+ if (j-i > 2)
+ C -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 2);
+ Matrix<Scalar,2,2> X;
+ matrix_sqrt_quasi_triangular_solve_auxiliary_equation(X, A, B, C);
+ sqrtT.template block<2,2>(i,j) = X;
+}
+
+// pre: T is quasi-upper-triangular and sqrtT is a zero matrix of the same size
+// post: the diagonal blocks of sqrtT are the square roots of the diagonal blocks of T
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular_diagonal(const MatrixType& T, ResultType& sqrtT)
+{
+ using std::sqrt;
+ const Index size = T.rows();
+ for (Index i = 0; i < size; i++) {
+ if (i == size - 1 || T.coeff(i+1, i) == 0) {
+ eigen_assert(T(i,i) >= 0);
+ sqrtT.coeffRef(i,i) = sqrt(T.coeff(i,i));
+ }
+ else {
+ matrix_sqrt_quasi_triangular_2x2_diagonal_block(T, i, sqrtT);
+ ++i;
+ }
+ }
+}
+
+// pre: T is quasi-upper-triangular and diagonal blocks of sqrtT are square root of diagonal blocks of T.
+// post: sqrtT is the square root of T.
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular_off_diagonal(const MatrixType& T, ResultType& sqrtT)
+{
+ const Index size = T.rows();
+ for (Index j = 1; j < size; j++) {
+ if (T.coeff(j, j-1) != 0) // if T(j-1:j, j-1:j) is a 2-by-2 block
+ continue;
+ for (Index i = j-1; i >= 0; i--) {
+ if (i > 0 && T.coeff(i, i-1) != 0) // if T(i-1:i, i-1:i) is a 2-by-2 block
+ continue;
+ bool iBlockIs2x2 = (i < size - 1) && (T.coeff(i+1, i) != 0);
+ bool jBlockIs2x2 = (j < size - 1) && (T.coeff(j+1, j) != 0);
+ if (iBlockIs2x2 && jBlockIs2x2)
+ matrix_sqrt_quasi_triangular_2x2_off_diagonal_block(T, i, j, sqrtT);
+ else if (iBlockIs2x2 && !jBlockIs2x2)
+ matrix_sqrt_quasi_triangular_2x1_off_diagonal_block(T, i, j, sqrtT);
+ else if (!iBlockIs2x2 && jBlockIs2x2)
+ matrix_sqrt_quasi_triangular_1x2_off_diagonal_block(T, i, j, sqrtT);
+ else if (!iBlockIs2x2 && !jBlockIs2x2)
+ matrix_sqrt_quasi_triangular_1x1_off_diagonal_block(T, i, j, sqrtT);
+ }
+ }
+}
+
+} // end of namespace internal
/** \ingroup MatrixFunctions_Module
- * \brief Class for computing matrix square roots of upper triangular matrices.
- * \tparam MatrixType type of the argument of the matrix square root,
- * expected to be an instantiation of the Matrix class template.
+ * \brief Compute matrix square root of quasi-triangular matrix.
*
- * This class computes the square root of the upper triangular matrix
- * stored in the upper triangular part (including the diagonal) of
- * the matrix passed to the constructor.
+ * \tparam MatrixType type of \p arg, the argument of matrix square root,
+ * expected to be an instantiation of the Matrix class template.
+ * \tparam ResultType type of \p result, where result is to be stored.
+ * \param[in] arg argument of matrix square root.
+ * \param[out] result matrix square root of upper Hessenberg part of \p arg.
+ *
+ * This function computes the square root of the upper quasi-triangular matrix stored in the upper
+ * Hessenberg part of \p arg. Only the upper Hessenberg part of \p result is updated, the rest is
+ * not touched. See MatrixBase::sqrt() for details on how this computation is implemented.
*
* \sa MatrixSquareRoot, MatrixSquareRootQuasiTriangular
*/
-template <typename MatrixType>
-class MatrixSquareRootTriangular
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_quasi_triangular(const MatrixType &arg, ResultType &result)
{
- public:
- MatrixSquareRootTriangular(const MatrixType& A)
- : m_A(A)
- {
- eigen_assert(A.rows() == A.cols());
- }
+ eigen_assert(arg.rows() == arg.cols());
+ result.resize(arg.rows(), arg.cols());
+ internal::matrix_sqrt_quasi_triangular_diagonal(arg, result);
+ internal::matrix_sqrt_quasi_triangular_off_diagonal(arg, result);
+}
- /** \brief Compute the matrix square root
- *
- * \param[out] result square root of \p A, as specified in the constructor.
- *
- * Only the upper triangular part (including the diagonal) of
- * \p result is updated, the rest is not touched. See
- * MatrixBase::sqrt() for details on how this computation is
- * implemented.
- */
- template <typename ResultType> void compute(ResultType &result);
- private:
- const MatrixType& m_A;
-};
-
-template <typename MatrixType>
-template <typename ResultType>
-void MatrixSquareRootTriangular<MatrixType>::compute(ResultType &result)
+/** \ingroup MatrixFunctions_Module
+ * \brief Compute matrix square root of triangular matrix.
+ *
+ * \tparam MatrixType type of \p arg, the argument of matrix square root,
+ * expected to be an instantiation of the Matrix class template.
+ * \tparam ResultType type of \p result, where result is to be stored.
+ * \param[in] arg argument of matrix square root.
+ * \param[out] result matrix square root of upper triangular part of \p arg.
+ *
+ * Only the upper triangular part (including the diagonal) of \p result is updated, the rest is not
+ * touched. See MatrixBase::sqrt() for details on how this computation is implemented.
+ *
+ * \sa MatrixSquareRoot, MatrixSquareRootQuasiTriangular
+ */
+template <typename MatrixType, typename ResultType>
+void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
{
using std::sqrt;
-
- // Compute square root of m_A and store it in upper triangular part of result
- // This uses that the square root of triangular matrices can be computed directly.
- result.resize(m_A.rows(), m_A.cols());
- typedef typename MatrixType::Index Index;
- for (Index i = 0; i < m_A.rows(); i++) {
- result.coeffRef(i,i) = sqrt(m_A.coeff(i,i));
- }
- for (Index j = 1; j < m_A.cols(); j++) {
- for (Index i = j-1; i >= 0; i--) {
typedef typename MatrixType::Scalar Scalar;
+
+ eigen_assert(arg.rows() == arg.cols());
+
+ // Compute square root of arg and store it in upper triangular part of result
+ // This uses that the square root of triangular matrices can be computed directly.
+ result.resize(arg.rows(), arg.cols());
+ for (Index i = 0; i < arg.rows(); i++) {
+ result.coeffRef(i,i) = sqrt(arg.coeff(i,i));
+ }
+ for (Index j = 1; j < arg.cols(); j++) {
+ for (Index i = j-1; i >= 0; i--) {
// if i = j-1, then segment has length 0 so tmp = 0
Scalar tmp = (result.row(i).segment(i+1,j-i-1) * result.col(j).segment(i+1,j-i-1)).value();
// denominator may be zero if original matrix is singular
- result.coeffRef(i,j) = (m_A.coeff(i,j) - tmp) / (result.coeff(i,i) + result.coeff(j,j));
+ result.coeffRef(i,j) = (arg.coeff(i,j) - tmp) / (result.coeff(i,i) + result.coeff(j,j));
}
}
}
+namespace internal {
+
/** \ingroup MatrixFunctions_Module
- * \brief Class for computing matrix square roots of general matrices.
+ * \brief Helper struct for computing matrix square roots of general matrices.
* \tparam MatrixType type of the argument of the matrix square root,
* expected to be an instantiation of the Matrix class template.
*
* \sa MatrixSquareRootTriangular, MatrixSquareRootQuasiTriangular, MatrixBase::sqrt()
*/
template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
-class MatrixSquareRoot
+struct matrix_sqrt_compute
{
- public:
-
- /** \brief Constructor.
- *
- * \param[in] A matrix whose square root is to be computed.
- *
- * The class stores a reference to \p A, so it should not be
- * changed (or destroyed) before compute() is called.
- */
- MatrixSquareRoot(const MatrixType& A);
-
- /** \brief Compute the matrix square root
- *
- * \param[out] result square root of \p A, as specified in the constructor.
- *
- * See MatrixBase::sqrt() for details on how this computation is
- * implemented.
- */
- template <typename ResultType> void compute(ResultType &result);
+ /** \brief Compute the matrix square root
+ *
+ * \param[in] arg matrix whose square root is to be computed.
+ * \param[out] result square root of \p arg.
+ *
+ * See MatrixBase::sqrt() for details on how this computation is implemented.
+ */
+ template <typename ResultType> static void run(const MatrixType &arg, ResultType &result);
};
// ********** Partial specialization for real matrices **********
template <typename MatrixType>
-class MatrixSquareRoot<MatrixType, 0>
+struct matrix_sqrt_compute<MatrixType, 0>
{
- public:
+ template <typename ResultType>
+ static void run(const MatrixType &arg, ResultType &result)
+ {
+ eigen_assert(arg.rows() == arg.cols());
- MatrixSquareRoot(const MatrixType& A)
- : m_A(A)
- {
- eigen_assert(A.rows() == A.cols());
- }
-
- template <typename ResultType> void compute(ResultType &result)
- {
- // Compute Schur decomposition of m_A
- const RealSchur<MatrixType> schurOfA(m_A);
- const MatrixType& T = schurOfA.matrixT();
- const MatrixType& U = schurOfA.matrixU();
+ // Compute Schur decomposition of arg
+ const RealSchur<MatrixType> schurOfA(arg);
+ const MatrixType& T = schurOfA.matrixT();
+ const MatrixType& U = schurOfA.matrixU();
- // Compute square root of T
- MatrixType sqrtT = MatrixType::Zero(m_A.rows(), m_A.cols());
- MatrixSquareRootQuasiTriangular<MatrixType>(T).compute(sqrtT);
+ // Compute square root of T
+ MatrixType sqrtT = MatrixType::Zero(arg.rows(), arg.cols());
+ matrix_sqrt_quasi_triangular(T, sqrtT);
- // Compute square root of m_A
- result = U * sqrtT * U.adjoint();
- }
-
- private:
- const MatrixType& m_A;
+ // Compute square root of arg
+ result = U * sqrtT * U.adjoint();
+ }
};
// ********** Partial specialization for complex matrices **********
template <typename MatrixType>
-class MatrixSquareRoot<MatrixType, 1>
+struct matrix_sqrt_compute<MatrixType, 1>
{
- public:
+ template <typename ResultType>
+ static void run(const MatrixType &arg, ResultType &result)
+ {
+ eigen_assert(arg.rows() == arg.cols());
- MatrixSquareRoot(const MatrixType& A)
- : m_A(A)
- {
- eigen_assert(A.rows() == A.cols());
- }
-
- template <typename ResultType> void compute(ResultType &result)
- {
- // Compute Schur decomposition of m_A
- const ComplexSchur<MatrixType> schurOfA(m_A);
- const MatrixType& T = schurOfA.matrixT();
- const MatrixType& U = schurOfA.matrixU();
+ // Compute Schur decomposition of arg
+ const ComplexSchur<MatrixType> schurOfA(arg);
+ const MatrixType& T = schurOfA.matrixT();
+ const MatrixType& U = schurOfA.matrixU();
- // Compute square root of T
- MatrixType sqrtT;
- MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
+ // Compute square root of T
+ MatrixType sqrtT;
+ matrix_sqrt_triangular(T, sqrtT);
- // Compute square root of m_A
- result = U * (sqrtT.template triangularView<Upper>() * U.adjoint());
- }
-
- private:
- const MatrixType& m_A;
+ // Compute square root of arg
+ result = U * (sqrtT.template triangularView<Upper>() * U.adjoint());
+ }
};
+} // end namespace internal
/** \ingroup MatrixFunctions_Module
*
@@ -415,14 +314,16 @@
template<typename Derived> class MatrixSquareRootReturnValue
: public ReturnByValue<MatrixSquareRootReturnValue<Derived> >
{
- typedef typename Derived::Index Index;
+ protected:
+ typedef typename internal::ref_selector<Derived>::type DerivedNested;
+
public:
/** \brief Constructor.
*
* \param[in] src %Matrix (expression) forming the argument of the
* matrix square root.
*/
- MatrixSquareRootReturnValue(const Derived& src) : m_src(src) { }
+ explicit MatrixSquareRootReturnValue(const Derived& src) : m_src(src) { }
/** \brief Compute the matrix square root.
*
@@ -432,18 +333,17 @@
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
- const typename Derived::PlainObject srcEvaluated = m_src.eval();
- MatrixSquareRoot<typename Derived::PlainObject> me(srcEvaluated);
- me.compute(result);
+ typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
+ typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
+ DerivedEvalType tmp(m_src);
+ internal::matrix_sqrt_compute<DerivedEvalTypeClean>::run(tmp, result);
}
Index rows() const { return m_src.rows(); }
Index cols() const { return m_src.cols(); }
protected:
- const Derived& m_src;
- private:
- MatrixSquareRootReturnValue& operator=(const MatrixSquareRootReturnValue&);
+ const DerivedNested m_src;
};
namespace internal {
diff --git a/unsupported/Eigen/src/MatrixFunctions/StemFunction.h b/unsupported/Eigen/src/MatrixFunctions/StemFunction.h
index 724e55c..7604df9 100644
--- a/unsupported/Eigen/src/MatrixFunctions/StemFunction.h
+++ b/unsupported/Eigen/src/MatrixFunctions/StemFunction.h
@@ -1,7 +1,7 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
-// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2010, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
@@ -12,93 +12,105 @@
namespace Eigen {
-/** \ingroup MatrixFunctions_Module
- * \brief Stem functions corresponding to standard mathematical functions.
- */
+namespace internal {
+
+/** \brief The exponential function (and its derivatives). */
template <typename Scalar>
-class StdStemFunctions
+Scalar stem_function_exp(Scalar x, int)
{
- public:
+ using std::exp;
+ return exp(x);
+}
- /** \brief The exponential function (and its derivatives). */
- static Scalar exp(Scalar x, int)
- {
- return std::exp(x);
- }
+/** \brief Cosine (and its derivatives). */
+template <typename Scalar>
+Scalar stem_function_cos(Scalar x, int n)
+{
+ using std::cos;
+ using std::sin;
+ Scalar res;
- /** \brief Cosine (and its derivatives). */
- static Scalar cos(Scalar x, int n)
- {
- Scalar res;
- switch (n % 4) {
- case 0:
- res = std::cos(x);
- break;
- case 1:
- res = -std::sin(x);
- break;
- case 2:
- res = -std::cos(x);
- break;
- case 3:
- res = std::sin(x);
- break;
- }
- return res;
- }
+ switch (n % 4) {
+ case 0:
+ res = std::cos(x);
+ break;
+ case 1:
+ res = -std::sin(x);
+ break;
+ case 2:
+ res = -std::cos(x);
+ break;
+ case 3:
+ res = std::sin(x);
+ break;
+ }
+ return res;
+}
- /** \brief Sine (and its derivatives). */
- static Scalar sin(Scalar x, int n)
- {
- Scalar res;
- switch (n % 4) {
- case 0:
- res = std::sin(x);
- break;
- case 1:
- res = std::cos(x);
- break;
- case 2:
- res = -std::sin(x);
- break;
- case 3:
- res = -std::cos(x);
- break;
- }
- return res;
- }
+/** \brief Sine (and its derivatives). */
+template <typename Scalar>
+Scalar stem_function_sin(Scalar x, int n)
+{
+ using std::cos;
+ using std::sin;
+ Scalar res;
- /** \brief Hyperbolic cosine (and its derivatives). */
- static Scalar cosh(Scalar x, int n)
- {
- Scalar res;
- switch (n % 2) {
- case 0:
- res = std::cosh(x);
- break;
- case 1:
- res = std::sinh(x);
- break;
- }
- return res;
- }
+ switch (n % 4) {
+ case 0:
+ res = std::sin(x);
+ break;
+ case 1:
+ res = std::cos(x);
+ break;
+ case 2:
+ res = -std::sin(x);
+ break;
+ case 3:
+ res = -std::cos(x);
+ break;
+ }
+ return res;
+}
+
+/** \brief Hyperbolic cosine (and its derivatives). */
+template <typename Scalar>
+Scalar stem_function_cosh(Scalar x, int n)
+{
+ using std::cosh;
+ using std::sinh;
+ Scalar res;
+
+ switch (n % 2) {
+ case 0:
+ res = std::cosh(x);
+ break;
+ case 1:
+ res = std::sinh(x);
+ break;
+ }
+ return res;
+}
- /** \brief Hyperbolic sine (and its derivatives). */
- static Scalar sinh(Scalar x, int n)
- {
- Scalar res;
- switch (n % 2) {
- case 0:
- res = std::sinh(x);
- break;
- case 1:
- res = std::cosh(x);
- break;
- }
- return res;
- }
+/** \brief Hyperbolic sine (and its derivatives). */
+template <typename Scalar>
+Scalar stem_function_sinh(Scalar x, int n)
+{
+ using std::cosh;
+ using std::sinh;
+ Scalar res;
+
+ switch (n % 2) {
+ case 0:
+ res = std::sinh(x);
+ break;
+ case 1:
+ res = std::cosh(x);
+ break;
+ }
+ return res;
+}
-}; // end of class StdStemFunctions
+} // end namespace internal
} // end namespace Eigen