Squashed 'third_party/eigen/' content from commit 61d72f6
Change-Id: Iccc90fa0b55ab44037f018046d2fcffd90d9d025
git-subtree-dir: third_party/eigen
git-subtree-split: 61d72f6383cfa842868c53e30e087b0258177257
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
new file mode 100644
index 0000000..78a307e
--- /dev/null
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
@@ -0,0 +1,508 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2012, 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
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_MATRIX_POWER
+#define EIGEN_MATRIX_POWER
+
+namespace Eigen {
+
+template<typename MatrixType> class MatrixPower;
+
+template<typename MatrixType>
+class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
+{
+ public:
+ typedef typename MatrixType::RealScalar RealScalar;
+ typedef typename MatrixType::Index Index;
+
+ MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
+ { }
+
+ template<typename ResultType>
+ inline void evalTo(ResultType& res) const
+ { m_pow.compute(res, m_p); }
+
+ Index rows() const { return m_pow.rows(); }
+ Index cols() const { return m_pow.cols(); }
+
+ private:
+ MatrixPower<MatrixType>& m_pow;
+ const RealScalar m_p;
+ MatrixPowerRetval& operator=(const MatrixPowerRetval&);
+};
+
+template<typename MatrixType>
+class MatrixPowerAtomic
+{
+ private:
+ enum {
+ RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+ MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
+ };
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename MatrixType::RealScalar RealScalar;
+ typedef std::complex<RealScalar> ComplexScalar;
+ typedef typename MatrixType::Index Index;
+ typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;
+
+ 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;
+ static int getPadeDegree(float normIminusT);
+ static int getPadeDegree(double normIminusT);
+ static int getPadeDegree(long double normIminusT);
+ static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
+ static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
+
+ public:
+ MatrixPowerAtomic(const MatrixType& T, RealScalar p);
+ void compute(MatrixType& 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()); }
+
+template<typename MatrixType>
+void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
+{
+ res.resizeLike(m_A);
+ switch (m_A.rows()) {
+ case 0:
+ break;
+ case 1:
+ res(0,0) = std::pow(m_A(0,0), m_p);
+ break;
+ case 2:
+ compute2x2(res, m_p);
+ break;
+ default:
+ computeBig(res);
+ }
+}
+
+template<typename MatrixType>
+void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const
+{
+ int i = degree<<1;
+ res = (m_p-degree) / ((i-1)<<1) * 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();
+ }
+ 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
+{
+ 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) {
+ res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);
+ if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))
+ res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);
+ else if (2*abs(m_A.coeff(i-1,i-1)) < abs(m_A.coeff(i,i)) || 2*abs(m_A.coeff(i,i)) < abs(m_A.coeff(i-1,i-1)))
+ res.coeffRef(i-1,i) = (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_A.coeff(i,i)-m_A.coeff(i-1,i-1));
+ else
+ res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);
+ res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);
+ }
+}
+
+template<typename MatrixType>
+void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
+{
+ 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
+ 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));
+
+ while (true) {
+ IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
+ normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
+ if (normIminusT < maxNormForPade) {
+ degree = getPadeDegree(normIminusT);
+ degree2 = getPadeDegree(normIminusT/2);
+ if (degree - degree2 <= 1 || hasExtraSquareRoot)
+ break;
+ hasExtraSquareRoot = true;
+ }
+ MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
+ T = sqrtT.template triangularView<Upper>();
+ ++numberOfSquareRoots;
+ }
+ computePade(degree, IminusT, res);
+
+ for (; numberOfSquareRoots; --numberOfSquareRoots) {
+ compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));
+ res = res.template triangularView<Upper>() * res;
+ }
+ compute2x2(res, m_p);
+}
+
+template<typename MatrixType>
+inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
+{
+ const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
+ int degree = 3;
+ for (; degree <= 4; ++degree)
+ if (normIminusT <= maxNormForPade[degree - 3])
+ break;
+ return degree;
+}
+
+template<typename MatrixType>
+inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
+{
+ const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
+ 1.999045567181744e-1, 2.789358995219730e-1 };
+ int degree = 3;
+ for (; degree <= 7; ++degree)
+ if (normIminusT <= maxNormForPade[degree - 3])
+ break;
+ return degree;
+}
+
+template<typename MatrixType>
+inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
+{
+#if LDBL_MANT_DIG == 53
+ const int maxPadeDegree = 7;
+ const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
+ 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,
+ 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
+#elif LDBL_MANT_DIG <= 106
+ const int maxPadeDegree = 10;
+ const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
+ 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
+ 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
+ 1.1016843812851143391275867258512e-1L };
+#else
+ const int maxPadeDegree = 10;
+ const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
+ 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
+ 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
+ 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
+ 9.134603732914548552537150753385375e-2L };
+#endif
+ int degree = 3;
+ for (; degree <= maxPadeDegree; ++degree)
+ if (normIminusT <= maxNormForPade[degree - 3])
+ break;
+ return degree;
+}
+
+template<typename MatrixType>
+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);
+}
+
+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);
+}
+
+/**
+ * \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 real/complex matrices raised to
+ * an arbitrary real power. Meanwhile, it saves the result of Schur
+ * decomposition if an non-integral power has even been calculated.
+ * Therefore, if you want to compute multiple (>= 2) matrix powers
+ * for the same matrix, using the class directly is more efficient than
+ * calling MatrixBase::pow().
+ *
+ * Example:
+ * \include MatrixPower_optimal.cpp
+ * Output: \verbinclude MatrixPower_optimal.out
+ */
+template<typename MatrixType>
+class MatrixPower
+{
+ 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;
+
+ public:
+ /**
+ * \brief Constructor.
+ *
+ * \param[in] A the base of the matrix power.
+ *
+ * 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)
+ { eigen_assert(A.rows() == A.cols()); }
+
+ /**
+ * \brief Returns the matrix power.
+ *
+ * \param[in] p exponent, a real scalar.
+ * \return The expression \f$ A^p \f$, where A is specified in the
+ * constructor.
+ */
+ const MatrixPowerRetval<MatrixType> operator()(RealScalar p)
+ { return MatrixPowerRetval<MatrixType>(*this, p); }
+
+ /**
+ * \brief Compute the matrix power.
+ *
+ * \param[in] p exponent, a real scalar.
+ * \param[out] res \f$ A^p \f$ where A is specified in the
+ * constructor.
+ */
+ template<typename ResultType>
+ void compute(ResultType& res, RealScalar p);
+
+ Index rows() const { return m_A.rows(); }
+ Index cols() const { return m_A.cols(); }
+
+ private:
+ typedef std::complex<RealScalar> ComplexScalar;
+ typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,
+ MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;
+
+ typename MatrixType::Nested m_A;
+ MatrixType m_tmp;
+ ComplexMatrix m_T, m_U, m_fT;
+ RealScalar m_conditionNumber;
+
+ RealScalar modfAndInit(RealScalar, RealScalar*);
+
+ template<typename ResultType>
+ void computeIntPower(ResultType&, RealScalar);
+
+ template<typename ResultType>
+ void computeFracPower(ResultType&, RealScalar);
+
+ template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
+ static void revertSchur(
+ Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
+ const ComplexMatrix& T,
+ const ComplexMatrix& U);
+
+ template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
+ static void revertSchur(
+ Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
+ const ComplexMatrix& T,
+ const ComplexMatrix& U);
+};
+
+template<typename MatrixType>
+template<typename ResultType>
+void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
+{
+ switch (cols()) {
+ case 0:
+ break;
+ case 1:
+ res(0,0) = std::pow(m_A.coeff(0,0), p);
+ break;
+ default:
+ RealScalar intpart, x = modfAndInit(p, &intpart);
+ computeIntPower(res, intpart);
+ computeFracPower(res, x);
+ }
+}
+
+template<typename MatrixType>
+typename MatrixPower<MatrixType>::RealScalar
+MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
+{
+ typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;
+
+ *intpart = std::floor(x);
+ RealScalar res = x - *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();
+ }
+
+ if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
+ --res;
+ ++*intpart;
+ }
+ return res;
+}
+
+template<typename MatrixType>
+template<typename ResultType>
+void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
+{
+ RealScalar pp = std::abs(p);
+
+ 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)
+ res = m_tmp * res;
+ m_tmp *= m_tmp;
+ pp /= 2;
+ }
+}
+
+template<typename MatrixType>
+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;
+ }
+}
+
+template<typename MatrixType>
+template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
+inline void MatrixPower<MatrixType>::revertSchur(
+ Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
+ const ComplexMatrix& T,
+ const ComplexMatrix& U)
+{ res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
+
+template<typename MatrixType>
+template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
+inline void MatrixPower<MatrixType>::revertSchur(
+ Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
+ const ComplexMatrix& T,
+ const ComplexMatrix& U)
+{ res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
+
+/**
+ * \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 MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
+{
+ public:
+ typedef typename Derived::PlainObject PlainObject;
+ typedef typename Derived::RealScalar RealScalar;
+ typedef typename Derived::Index Index;
+
+ /**
+ * \brief Constructor.
+ *
+ * \param[in] A %Matrix (expression), the base of the matrix power.
+ * \param[in] p scalar, the exponent of the matrix power.
+ */
+ MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
+ { }
+
+ /**
+ * \brief Compute the matrix power.
+ *
+ * \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& res) const
+ { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }
+
+ Index rows() const { return m_A.rows(); }
+ Index cols() const { return m_A.cols(); }
+
+ private:
+ const Derived& m_A;
+ const RealScalar m_p;
+ MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
+};
+
+namespace internal {
+
+template<typename MatrixPowerType>
+struct traits< MatrixPowerRetval<MatrixPowerType> >
+{ typedef typename MatrixPowerType::PlainObject ReturnType; };
+
+template<typename Derived>
+struct traits< MatrixPowerReturnValue<Derived> >
+{ typedef typename Derived::PlainObject ReturnType; };
+
+}
+
+template<typename Derived>
+const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
+{ return MatrixPowerReturnValue<Derived>(derived(), p); }
+
+} // namespace Eigen
+
+#endif // EIGEN_MATRIX_POWER